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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�
arXiv:cond-mat/0209131v1 [cond-mat.stat-mech] 5 Sep 2002
Steven R. Pridea,
∗
, Renaud Toussaintb
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 6090Email addresses: Steve.Prideuniv-rennes1.fr (Steven R.Pride),
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 freeto displa e. A load
F
N
is applied to the bundle through the free support so thatthe bers are inastateof tension.Inthis paper, theloadF
N
willalways benormalizedbyN
todenetheoverall tensionτ = F
N
/N
.Ea hnon-broken0000000000000000000000000000000
0000000000000000000000000000000
0000000000000000000000000000000
0000000000000000000000000000000
0000000000000000000000000000000
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 loadF
N
applied throughthefree support.berinabundlehasthe samelength
L
.If,whenτ = 0
,thislengthisL
o
,then the measure of strainisε = L/L
o
− 1
.Experimentsmay be performedonthe bundleeither by ontrollingτ
orε
.Ea hofthebershasthe sameYoung'smoduluswhi histaken tobeunityso
thatthe axialstrain
ε
of ea hberisidenti al tothetensioninthe ber.TheN
bers have strengthsε
1
,ε
2
,...ε
N
whi h are independent random variables sampled from a distributionp(ε)
, whose umulative distribution is denedP (ε) =
R
ε
0
p(x)dx
. As the strainε
of the bundle is in reased, the individual bers willbreak on etheirtension(strain)getstotheir xedstrengththresh-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 statej
of damage when the ensemble as a whole is at an applied strainε
. In the ber bundle model, a damage statej
is dened by whi h of theN
bers are broken. One ould denej
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 statej
is obtained by maximizingShannon's measure of disorderS = −
X
j
p
j
ln p
j
(1)subje tto onstraints.Su h onstraintsmustinvolvetheindependentvariables
of
S
. To identify the independent variables, we onsider how bothS
and the average energy inthe ensembleof bundles hange asthe strainis in reased.When
ε
in reases toε + dε
, there is both a work arried out in reversibly stret hing the bers and an additional work arried out due to irreversiblerealizations of the disorder) willbe ledout of their urrent damagestate and
into state
j
, while others that were in statej
will transfer to still dierent states. Ifthere is adieren e inthe numberof members entering and leavingstate
j
, therewillbea hangedp
j
inthe o upationprobabilityof statej
and su h hangesare what ause Shannon'sdisorder measureS
to hange.Theaverageenergydensity(averageenergynormalizedby
N
)intheensemble isgivenbyU =
P
j
p
j
E
j
.Here,E
j
istheenergydensityrequiredto reatestatej
at imposed strainε
and as averaged over all members that have been led to statej
. Depending on the breaking strengths of a given realization, the work performed in arriving at statej
an be dierent. It is throughE
j
that alldependen e onthequen hed-disorderdistributionentersthe problem.Thehange that o urs when
ε
in reases toε + dε
isdU =
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 toε+dε
,thereisaowofmembersbetween thestates.Thisowinvolvesenergy hanges due to bers breaking in the intervaldε
. It may be resolved into a uniformin ompressible part whoseasso iatedenergy isin ludedwithinf dε
as well as a non-uniform ompressible part asso iated with more membersarriving 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,thenU = U(S, ε)
,orequivalentlyifS
istobetreatedas the fundamentalfun tion thenS = S(U, ε)
. In other words, the independent variables that must be involved in the onstraints on the maximizationofS
are
U
andε
. The proportionality onstantsT
andf
are denedT =
∂U
∂S
!
ε
andf =
∂U
∂ε
!
S
.
(5)Thestatefun tion
f
issomethingdierentthantheoverall tensionτ
sin e we alsohavethatτ dε = dU
.Thus,ingeneral,(τ −f)dε = T dS
sothatf 6= τ
due to bers breaking in a positive in rementdε
. If strain were to be de reased, bers do not break and sodS = 0
and the state fun tionf
would be dened using onlythe purely elasti part of the energy hangesdE
j
. Changes inf
in this ase are equivalent to hanges inτ
. Last, sin e we have takenS
to be extensive(proportionaltoN
)whileU
isadensityindependentofN
,T
omes out being proportional to1/N
. This hoi e is made so that fa tors ofN
do not lutter the equationsthat follow.The onstraintinvolving
ε
isthat ea hnon-brokenberthroughouttheentire ensemble has the same length whi h impliesε
j
= ε
. The onstraint involvingU
is thatU =
P
j
p
j
E
j
. Carrying out the maximizationofS
subje t tothese onstraintsusing Lagrangianmultipliersgivesthe probabilitydistributionasp
j
(β, ε) =
e
−βE
j
(ε)
Z(β, ε)
(6)where
β = 1/T
and where the partitionfun tionZ
isdenedZ(β, ε) =
X
j
e
−βE
j
(ε)
.
(7)
Thefreeenergy
F = F (β, ε)
inthisensemble(forthisset of onstraints)isthe Legendre transformofU
withrespe t toS
; i.e.,F = U − T S
. Dierentiation then givesdF = β
−2
Sdβ + f dε.
(8) Uponintrodu ingU =
P
j
p
j
E
j
andS = −
P
j
p
j
ln p
j
intotheLegendre trans-formF = U − S/β
andusingthatln p
j
= −βE
j
− ln Z
weobtainthestandard resultβF = − 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 fromthe previously stated but unused fa t that
dU = τ dε
whi h an bewrittenτ =
dU
dε
=
X
j
dp
j
dε
E
j
+
X
j
p
j
dE
j
dε
.
(10)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 loadτ
mustberedu ed asrepresentedbytheverti al drops. Now fromEq. (6), we obtain thatdp
j
dε
=
"
−
dE
dε
j
β − E
j
dβ
dε
−
d ln Z
dε
#
p
j
,
(11)while fromEqs. (8)and (9)
d ln Z
dε
= −F
dβ
dε
− β
dF
dε
= −U
dβ
dε
− βf.
(12)Sin e ea h member is at the same
ε
, ea h member has its ownτ
j
, and soτ =
P
j
p
j
τ
j
. We then obtain the dierential equationforβ
inthe forma
dβ
dε
+ 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
dε
!
; c =
X
j
p
j
dE
j
dε
− 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 tionβ = β(ε)
exa tly satisesthisequationand thusisthetrueber-bundleβ
.Tomakeprogress, wenext obtainE
j
andτ
j
forthespe i problemofberbundleswith global-load sharing.3 FIBER BUNDLE MODEL
Figure2detailsthehistoryofhowtheoveralltension
τ
mightevolvewhen on-trolledvariationsinε
areappliedtoabundlehavingberstrengthsε
1
, ε
2
...ε
N
. In this parti ular example,n = 3
bers have broken when the strain isatε
.Theloadonthe bundleisequallysharedbythe
N − n
survivingbers sothatτ
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 thresholdsε
1
, ε
2
...ε
n
.Thetotal workdensity
E
p
j
forstatesj
onsistingofn
broken bers isthe area under the harging urve in a parti ular realization (the area under the solidlinein 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-ventionε
n+1
= ε
is the nal applied strain. A dire t re ursion gives exa tlyE
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 statej
. Ea h breaking thresholdε
m
is an independantvariable,randomlydistributeda ording top(ε
m
)
underthe on-ditionthat0 ≤ ε
m
≤ ε
. They are thereforedistributed a ording tothe prob-ability densityp(ε
m
)/P (ε)
where the normalization fa tor a ounts for the fa t that the upper limitε
is independent of the threshold valuesε
m
. Thus, averagingover the quen hed disorder givesh(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 ingeneralE
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 thatp(ε) = 1
,P (ε) = ε
,h(ε) = ε
2
/6
,and
E
j
= ε
2
/2 − (n
We now apply the above theory and determine the thermodynami fun tions
asafun tionofappliedstrain
ε
.Thefollowinganalysisisvalidforanyproperly normalized quen hed-disorder distributionp(ε)
.The umulativedistribution
P (ε) =
R
ε
0
p(x) dx
is the probabilitythatany one ber has broken when the strain is atε
. Thus, the fra tion of all possible realizations havingn
j
broken bers andN − n
j
unbroken bers is exa tlyp
exa tj
= (1 − P )
N −n
j
P
n
j
.
(19) This distribution may be writtenp
exa tj
= p
0
exp
−n
j
ln
1 − P
P
(20) wherep
0
= (1−P )
N
istheprobabilityoftheunbroken state
j = 0
.Uponusing the Hamiltonianof Eq. (18), the postulate of entropy maximizationpredi tsthis same distribution tobe given by
p
j
= p
0
exp
"
n
j
β
N
ε
2
2
− h
!#
(21) wherep
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 temperatureT = 1/β
is given byβ(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 hash(ε) < ε
2
/2
for any
distribution
p(ε)
.Thus,β(ε)
isanegative in reasingfun tionup tothe strain pointε = ε
β
whereitsmoothlygoestozero.Theine tionpointε
β
isobtained fromthe onditionthatP (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 tionofε
.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 squaredh(n
j
/N)
2
i
. Using the exa t
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) wherec
N
n
= N!/[n!(N − n)!]
denes the numberof ways of sele tingn
obje ts froma group ofN
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 multiplyingbyx/N
,givesthat whenx = P
andy = 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 tobeτ =
X
j
p
j
τ
j
= (1 − P )ε
(27)whi h is initiallyanin reasing fun tion of
ε
havingthe slopedτ
dε
= 1 − P − εp.
(28)This slope goes tozero and the stress has a maximum at any strain point
ε
τ
that is asolution of1 − P (ε
τ
) − ε
τ
p(ε
τ
) = 0
;i.e., atthe point(s) whereε
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 simultaneouslyεp(ε) = 1/2
andR
ε
0
p(x) dx = 1/2
whi h for monotoni distributionfun tionsp(ε)
anonlyo urwiththeuniformdistributionp(ε) =
1
in whi h aseε
τ
= ε
β
= 1/2
and the two transitions oin ide. Ifp(ε)
is a monotoni in reasing fun tion ofε
, thenε
τ
< ε
β
while if it is a de reasing fun tionofε
,thenε
τ
> ε
β
. Fornon-monotoni distributions, there an bean arbitrarynumberof stress maximasandEq. (29) an haveeithernosolutionsstrain points
ε
β
andε
τ
is dis ussed inthe following se tion. The average energy inthe ensembleisU =
X
j
p
j
E
j
= (1 − P )
ε
2
2
+ hP.
(30)Againre allingthe denition of
h
fromEq. (17) givesdU
dε
= (1 − P )ε − p
ε
2
2
+
d(hP )
dε
= τ
(31)whi hisalsothe equationthat givesthedierentialequationfortemperature.
Thisissu ientfordemonstratingthatthe
β(ε)
ofEq.(22)satisesits dier-entialequation. Nonetheless, asa onsisten y test, the oe ientsa
,b
, andc
dened inEq. (14) willbederived and the dierentialequation willexpli itlybe shown to besatised.
To obtainthe state fun tion
f
, we needrstdE
j
dε
= 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
dε
= (1 − P )ε + p
ε
2
2
− h
!
.
(33)The variation of the entropy isobtained fromthe energy balan e as
dS
dε
= β
dU
dε
− f
!
= Np ln
1 − P
P
.
(34)Togetherwiththe initial ondition
S(0) = 0
,thisisreadily integrated togiveS = −N [P ln P + (1 − P ) ln(1 − P )] .
(35) Thisexpression isthe lassi alShannonresultforasetofN
randomvariables in two possible states having probabilitiesP
and1 − P
, whi h is pre isely the ase of the ber bundle with global sharing. This is another onsisten yhe k. This entropy
S
in reases fromzero [total ertainty that every member isinta t℄andgoessmoothlythroughamaximum[totalun ertaintyastowhatstateamembermaybein℄atthesamestrainpoint
P (ε
β
) = 1/2
whereβ
goes tozero.Afterthe smoothmaximum,S
de reasestorea hzeroifP (ε)
rea hes1
in whi h ase there is total ertainty that ea h member is entirely broken. WenotethatEq.(35) analsobedire tlyobtainedfromtheShannonformulaFinally,the freeenergy
F
is again obtained fromitsLegendre transform de-nitionF = U − S/β
to beF =
ε
2
2
−
ε
2
2
− h
!
ln(1 − P )
ln(1 − P ) − ln P
.
(36) At the pointε
β
dened byP (ε
β
) = 1/2
, the free energy diverges due to the fa t thatβ(ε
β
) = 0
whileS(ε
β
)
remains nite. So long asε
β
6= ε
τ
, the free energy does not diverge when (if)ε = ε
τ
. In passing,we alsonote thatZ =
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 temperatureasa = −
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 derivativedβ
dε
= −
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 byP (ε
β
) = 1/2
is where simultaneouslyβ = 0
, the entropy is a maximum, and the free energy diverges. It is distin t fromthe stress-maximum transition(s) at
ε
τ
. The interpretation ofε = ε
β
as a phasetransitionis natural,sin e the most probable ongurationofa bundlesuddenly jumps from being entirely inta t to entirely broken. What are the
measurable manifestationsof this transitionat
ε
β
?Sin etheentropyisamaximumatthispoint,theu tuationsbetween
brokenbersinstate
j
tobeρ
j
= n
j
/N
,anddenetheaverageofthisfra tion to beρ = hρ
j
i = P
whereP = 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 experimentsonber 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 transitionpointε
β
. Re all that for mono-toni quen hed-disorder distributions, ifp(ε)
is a de reasing fun tion (more weak bers than strong bers), thenε
β
< ε
τ
. So the numeri al observation inthis ase should be that∆ρ
goesthrough a maximum prior topeak stress. Equivalent omments hold whenp(ε)
is anin reasing fun tion andε
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,inthelimitasε → ε
β
the singular part ofF
diverges a ording tothe s aling lawF
s
= −
ln 2
8
[ε
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 phaseThephasetransitionatpeakstress
ε = ε
τ
istheonethatresear hersuptonow havefo usedon.Sin eS
anditsderivativesremain ontinuousandnitethere, an attempt to lassify this transition as being rst-order or se ond-order ismeaningless.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 tionintheneighborhoodofitsmaximumoneisguaranteed
|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 theorya 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 asD(∆) ∼ ∆
−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.
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.
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