Fractal solid mechanics

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Revisión
RevistaMexicanadeFísica40,No.
4 (1994)506-53!
Fractalsolidmechanics
A.S.BALANKIN AND P.TAMAYO
Instituto Tecnológicoy deEstudios SuperioresdeMonterrey
CampusEstado deMéxico,México
Recibidoel 23deagostode1993;aceptadoellO demarzode1994
ABSTRAc:;T.Thepurposeof presentarticleistoreviewanewgeneralapproachtosalidmechanics,
named,fractalsalidmechanics.The attentionis focusedonsystematicaccountof theproposed
basicconcepts,aswellasonthemostimportantresultof fractalsalidmechanics.Specialattention
is paidto thethermodynamictheoryof elasticityof multifractalswhichis effectivefor modeling
varioustypesof behaviorpatternsof deformedmaterialswith multifractalmicrostructure.The
fractal conceptsin fracturemechanicsareconsidered.It is shown,that the natureof fractal
geometryof fractureof a salidis associatedwithfundamentalphenomenonof transversestrains
of salid(Poisson'seffect).This is manifestedbytheself-similarityof self-affinityof heterogeneous
stressesin irreversiblydeformedsolids.Someof themostusefulanalyticalandcomputermodels
arediscussed.The resultof theoreticalpredictionsarecomparedwith experimentaldata.It is
shownthat theproposedapproachis veryeffectivefor adequatedescriptionof variousbehavior
patternsandsomeotherphenomenain deformedsolids.
.
RESUMEN.El presentetrabajosefundamentaenel nuevométodoparala físicadel estadosólido
denominadamecánicafractal del sólido.Seacentúala atenciónde un análisissistemáticode
.
las concepcionesy desarrollode los principiosfundamentalesplanteadospor la teoríageneral
de fractalesy una seriede modelosno linealesaplicablesa materialesde elevadaelasticidad
(elastómeros),plasticidady fracturadematerialesconestructuramultifractal.Sedemuestraque
la naturalezadela geometríafractalsebasaenel fenómenodedeformacióntransversaldel sólido
(efectodePoisson).Estopuedeserexplicadoporlaafinidaddeloscamposdetensión-deformación,
cuandoel sólidopresentad~formaciónirreversible.Seanalizanunaseriedeimportantesmodelos
analíticosy computacionales.Losresultadosdelosplanteamientosteóricossecomparancondatos
experimentales.Sedemuestraqueel métodopropuestoesefectivoparadescribiry explicaruna
seriedecomportamientosy algunosfenómenosenel sólidodeformado,yel disímilcomportamiento
decuerposdeformables.
PACS:02.50;62.20.D;81.40
1.INTRODUCTION
Nowit is obviousthat theclassicalapproximationof homogeneouscontinuumis not
adequatefor real heterogeneoussolidsand,therefore,is not usableas a generalmodel.
In themostgeneralcase,naturalor artificialheterogeneousmaterialsconsistof domains
of differentmaterials(phases)or of thesamematerialin differentstates.Furthermore,
crystalsandpolycrystals,polymersandcomposites,rocksandporousmedia,gridsand
multibarsystemscanbeconsideredasmediawith microstructure[1-4].And finally,re-
centIy,it hasbeendeterminedexperimentallythat afterirreversibledeformationeven
initiallyhomogeneouscontinuoussolid exhibitshierarchicalblockstructure[4-6].
FRACTAL SOLID MECHANICS
507
A characteristicfeatureof all materialswith microstructureis theexistenceof scale
parametersLi whicharerelatedto microgeometryor long-rangeinteractingtorees.The
natureof hierarchicalblockstructureof deformedsalidis determinedbyits fundamental
property,namely,its resistanceto shear,that causesthedifferencebetweenthecharac-
teristicspatialscalesof theregionsof localizationanddissipationof energypumpedinto
a deformedbodyby anexternalaction[7,8].That is whyin recentyears,newphysical
and mathematicalmodelaof materialmedia,whichcanbe consideredas far-reaching
generalizationsof c1assicaltheoriesof elasticity,viscoelasticityandplasticityhavebeen
intenselydeveloped.TheCosseratcontinuum[9]washistoricallyoneof thefirstmodelaof
elasticmediawhichcouldnotbedescribedwithinthescopeofc1assicalelasticity.However,
theworksof E.andF.Cosserat(1909)remainedpracticallyunknownfor a longtime,
andonly around1960did thegeneralizedmodelaof theCosseratcontinuumbeganto
bedeveloped.Now,theyareknownasorientedmedia,micromorphic,multipolar,asym-
metric,couple-stress,etc.theories[1].Explicitor implicitnonlocalityis thecharacteristic
featureof all suchtheories.The latter,in its turn,displaysitself in that thetheories
containparameterswhichhavethedimensionof length.':!'hesescaleparameterscanhave
differentnaturesuchasdistancebetweenpartic1esindiscretestructures,thedimensionof
a grainor a cell,characteristicradiusof correlation,or actingat certaindistancetorees,
etc.[1-4].A.Griffithpioneeredtheuseofcharacteristicscaleparametersinthetheoriesof
plasticityandfractureof salida.Now,variouscharacteristicscaleparametersof length's
unit areusedin themostof moderntheoriesof irreversibledeformationsandfracture
of solids[2-4,10].However,thefoundationof thesetheories,as well as of couple-stress
theoriesof elasticity,is theeuc1ideangeometry.
At thesametime,in thelast tenyearsit hasbecomec1ear,dueto effortsof many
scientists(see,forexample,Refs.[4,11-15]),thatalargeamountof objectsandprocesses
in naturemostbecharacterizedby non-euc1ideangeometryandhavescalingbehavior.
Theconceptof fractalshasrl.¡{s;entlybecomepopularin naturalsciences.This conceptis
introducedonthebasisof Hausdorff-Besicovitchdimensionwhichmayexceedtopological
dimensionof theobjectandmayassumefractionalvalues.Theconceptof dimensionality,
investigatedactivelyby mathematiciansfromthebeginningof thepresentcentury,has
beenbroughtto the attentionof physicistsin the monographof B.Mandelbrot[11],
whichappearedin 1975in French,andthenin 1977in English.This bookcanbere-
gardedas an excellentexampleof scientificadvertisingof popularization,in this case
advertisingof newconceptsandmodela.This is,in particular,reflectedin identification
of specialc1assof objects,whicharecalledfractals,whosemetric(Hausdorff-Besicovitch)
dimensionis differentfromits topologicaldimension.Anotherfundamentalpropertyof
fractals,distinguishingthemfromhomogeneouseuc1ideanobjects,is their scalingin-
variance(self-similarity)[11-15].Fractalsfoundin nature,suchascolloidalaggregates,
aerogel,polymers,dendriticpartic1es,porousmedia,surfacesandspatialdistributions
of cracksin salida,fracturesurfaces,etc.,differfromregularfractals(Cantorset,Peana
andKochcurves,Sierpinskigasketandcarpet,etc.)becausetheyexhibitonlystatistical
self-similarityinalimitedrangeofspatialscalinglengthsLo
<L <LM (see,forexample,
Refs.[11-15]).Themajorityofthemcanberegardedasanensembleoffractalsof different
dimensionscharacterizedby differentweights.Suchobjectsarecalledmultifractalsand
havea spectrumof Rényi dimensionsDq with -00 < q < 00[15].The self-similarity
508
A.S.BALANKINANOP.TAMAYO
propertyof multifractalsis morecomplicatedthanthat of homogeneousfractalsandis
describedin termsof multiscaling[16].A recentthermodynamiealformulationof multi-
fractals,thatof Feigenbaumetal.[17],mapsthemultifractalmeasureintosomepopular
physiealmodels,suchasIsingmodel,etc.[18].
Topologyandgeometryof naturalobjects,rangingin sizefromatomicscaletothesize
of universeis centralto themodelswemakein arderto"understand"and"describe"
nature.The last decadehasseenrecognitionof theimportant,andsometimesdecisive,
influenceof thefractaltopologyon theelasticand fracturepropertiesof solids[4,19-
30].The unityof lawsgoverningdynamicsof nonlinearsystems,aswell asrelationships
betweenthedimensionsof phasepathsandLyapounovexponentsareamongtheobvious
reasonsfor thegrowinginterestin thedynamicsof fractalsandin thefractaldynamics
of deformedsolids.Amongtheseoneshouldmentíanpartieularlythetapiesof elasticity
of self-similar(statistieallyself-similar,in generalcase)structures,closelyrelatedto the
problemsof rubberelasticityof polymersand nonlinearelasticityof composites,and
problemsof fractalgeometryof fractureof solids.
Twofundamentallawsof reversibledeformationof statistieallyself-similarmultifractal
structureswereproposedin ourworks[19,20].A generalthermodynamiealtheoryof elas-
ticityof fractalsandmultifractalswasdevelopedinourworks[20,21]onthebasisof these
laws.It wasshownthat thetransversedeformationexponentof anelasticallyisotropic
multifractalis defineduniquelyby its metriedimension.Theoretiealpredictionsagree
wellwithexperimentaldata,andwiththeresultsof computersimulations.The modelof
irreversibledeformationsof solidswith multifractalmierostructurewasproposedin our
works[22,23].Lateronwehaveusedtheseconceptsforsolvingsomeproblemsof fracture
mechanics[21,24,25].It wasshownthattheappearanceoffractaltopologyduringfracture
ofsolidsistheconsequenceofPoisson'seffect.Someconclusionsconcerninginterpretations
of solutionsto theproblemsof lineartheoryelasticityandcrackproblemswereobtained
onthebasisof fractalconceptoVariousmodelswereproposedfordeterminationof fractal
geometry,dynamies,and fractalkinetiesof the destructionof solidsin Refs.[31-40].
Generalresultsof theseworksprovidethe basison whiehthe fractal salidmechanies
couldbedeveloped.
This reviewis focusedonsystematicaccountof thebasieconcepts,fundamentalprin-
cipIesandthemostimportantresultsof fractalsolid mechanies.AIsosomeof themost
usefulanalytiealandcomputermodelsof fractalsalidmechanicsarediscussed.
2.THE CONCEPT OF FRACTALS AND THE BASIS OF MULTIFRACTAL ANALYSIS OF DE-
FORMED SOLIDS
After theintroductionof fractalgeometryby B.Mandelbrotthe keyproblemwasto
understandwhynaturegivesrigeto fractalstructures.This impliestheformulationof
modelsof fractalgrowthbasedonphysicalphenomenaandsubsequentunderstandingof
theirmathematicalstructurein thesamesenseastherenormalizationgrouphasallowed
to understandIsingtypemodels.The modelsof the limiteddiffusionaggregationand
themoregeneraldielectricbreakdownmodel,basedoniterativeprocessesgovernedbya
Laplaceequationandstochastiefield,haveclearphysiealmeaningandtheyspontaneously
FRACTAL SOLID MECHANICS
509
evolveintorandomfractalstructuresof greatcomplexity.However,fromtheoreticalpoint
ofview,it isnotpossibletodescribethemwithintheframeworkof usualconcepts.Fractal
geometryandmultifractalanalysisaremostwidelyusedin statisticaltopography,phys-
ical kinetics,statisticalphysics,Huidmechanics,geology,geophysics,andastrophysics,
polymerphysics,andbecomeincreasinglymoreimportantin materialscienceandsalid
mechanics,includingfracturemechanics.Thesemodelsarealgousedinquantummechan-
ics,relativisticquantumtheory,electrodynamics,relativitytheory,etc.Methodsof fractal
geometryfindincreasingusein thedescriptionof microstructureof structuralmaterials,
especiallyof compositematerials.The mostadequatedescriptionof fracturesurfaces
canbegivenin termsof fractals.For thestudyof physicochemical,electrophysical,and
mechanicalpropertiesof disordered,disperse,andporousmaterialsdifferentfractaland
multifractalmodelsarealgowidelyused.
In recentyears,it hasbeenestablishedthat severalphysicalquantitiesdescribing
randomsystemsdo not obeyconventionalscalinglaws.Prominentexamplesarethe
probabilitydensityof randomwalkson randomfractals,suchas percolationclusters,
voltagedropsin randomresistornetworks,growthprobabilitiesof limiteddiffusionag-
gregation,growthdynamicsof viscousfingeringat highcapillarynumbers,distribution
of impuritieson the surfacesandin thebulk,invariantmeasureof strangeattractors
in chaoticdynamicalsystems,andsomephenomenain deformedmedia,specificallythe
energydissipationin turbulence,'clustersof microscopicdefectsin deformedsolids,cas-
cadesof debrisgeneratedbyexplosionsor shocks,etc.(seeRef.[4]).All thesequantities
haveverybroaddistribution,andtheirmomentscannotbedescribedby a singleexpo-
nentbut an infinitehierarchyof exponentsis neededto characterizethem.The point
is that this descriptionholdswhateverthelinearscaleis -be it global,on the scale
of meters,or on the microscopicscale.This phenomenonis calledmultifractalityand
wasfirst foundin thecontextof turbulence.Multifractalsarefractalsetswhicharenot
self-similar.Multifractalmeasurearerelatedto thestudyof distributionof physicalor
otherquantitieson a geometricsupport.The supportmaybe an ordinaryplane,the
surfaceof a sphereor of a volume,or it coulditselfbea fractal.Conceptsunderlying
recentdevelopmentof what is nowcalledmultifractalswereoriginallyintroducedby
Mandelbrotin thediscussionof tllrbulenceandextendedby Mandelbrot,Grassberger,
Aharony,Meaking,Anisimov,Feder,Balankin,HentschelandProcaccia,Grassbergerand
Procaccia,BadiiandPoliti,Arcangelisetal.,Rammaletal.,Halseyetal.,Feigenbaumet
al.,andotherauthorsto manyotherateas(seeRefs.[4,14,15]).Analysisof experimental
dataandintroductionof thedimensionfunction1(0:)by Frish andParisi and Jensen
et al.gavethemostremarkableagreementbetweeno~servationandsimpletheoretical
modelof multifractals.Theydemonstratedtheusefulnessof multifractalsin describing
experimentalobservations.Relatedworksaredescribedby Glazieret al.,Bensimonet
al.,Halseyet al.,Meakin et al.,and Nittmannet al.The connectionbetweenmulti-
fractalityand multiscalingwasinvestigatedby Coniglioand Zannetti,andVavrivand
Ryabov[15,16].It hasbeenproposedthat multifractalscanbecharacterizedby aninfi-
nite spectrumof generalizeddimensionsDq,where-00
<
q
<
-00,algocalledthe Rényi
dimensions[12-15].
In bis attemptto generalizetheconceptof entropyof a probabilitydistribution,ttle
HungarianmathematicianA.Rényi introducedthe followingexpression basedon the
510
A.S.BALANKINANDP.TAMAYO
momentsoí arderqoí theprobabilitiesp¡:
N
1q
=
(q-l)-lln 2:P!,
i=1
(1)
whereq is not necessarilyan integer.In the limit q
-.
1 this definitionbecomesthe
well-knowniníormationentropy
N
11
=-
2:
Pi InPi,
i=1
(2)
ofadiscreteprobabilitydistribution,alsocalledShannonentropy.ThedefinitioninEq.(1)
canthereíorebeconsidered,aswasRényi'sintent,asgeneralizedentropy.For definition
oí generalizeddimensionsDq,onemustchoosetheportionoí themultiíractalwithboxes
of sizer,anddefinePi(r) astheprobabilityof findingthepointof structurein thei-th
box.It'shouldbenotedthat
N(r)
2:
p¡(r)
=
1.
i=1
(3)
Theset¡.L={P¡.}is calledmultifractalprobabilitymeasureor maSsdensity,of thesubset
Pi(r) e F containedinsidethei-th coveringboxwiththeedger.Consequently,theRényi
dimensionof arderqis givenbytheexpression
{
}
{
N(r) q
}
Dq
=
lim 1q(r)
=
-(q -l)-llim
'"
Pi (r)
,
r-+O Inr r-+O L- Inr
i=1
(4)
wheretheparameterqrangesfrom-00 to oo.
Onecanreadilyseethat in generalcase,Doisequalto metricdimensionevaluatedby
meansoí box-countingalgorithm,andcalledfractaldimension
DF ==Do
=
DB.
Generalizeddimensionofarderq
=
1isequaltoinformationdimensionDI;anddimension
D2isequaltothecorrelationintegralexponentDe,alsocalledcorrelationdimension.Note
that,generalizedRényidimensionssatisfytherelation
Dql::;Dq,
for q'>q;
theequalitybeingobtainedin thecaseof uniformsets,i.e.,suchthat theprobability
measureis constant,
1
Pi ==N(r)'
FRACTAL SOLID MECHANICS
511
TABLE 1.Rényi dimensionDq,speciaIvaluesand limitsoí sequencesoí massexponentsr(q)
andf(a) curveíor a muItiíractalmeasureM
={Pi},supportedbyasetwithíractaldimension
DF
=
dimHM (dimHM
=
Hausdorff-Besicovitchdimension),íor different momentsoí arder q oí
M (seeReís.[12-14]).
*The measureM hasentropy1
=-
lim1(r)/limr
=fI,
which is the íractaI dimensionoí the set
oí concentrationforthemeasureM (herel(r) istheentropyoí partitionoí measureMover boxes
oísize (r)).
*~HereP+andP- aretheIargestandthesmalIestprobabilitiesin boxesoí sizer.
andthegeneralizeddimensionDqequalsthemetricdimensionDB
=
DHforall q.Specifi-
rally,foranhomogeneousobjectall dimensionsDqareequaltoitstopologicaldimension,
~.e.,".
Dq ==DH =dT;
and for self-similarfractalsthe definitionin Eq.(4) gives
dT
~
Dq ==Do
=
DB
~d
for all valuesof q.A relateddimensionfunctionD(r) was introducedby Badii and
Politi [15].It wasshownthatRényidimensionsarerelatedtothesequenceof massexpo-
nentsr(q),to thedimensionF(a) curve,andto thedimensionfunctionD(,) in general
waysthat is usefulin applications(seeRefs.[4,12]).The Rényi dimensionDq,special
valuesandlimitsof sequenceof massexponentsr(q),andf(a) curvefor a multifractal
supportedbyasetwithfractaldimensionDF,for difIerentmomentsof arderqaregiven
in Table1.Theseconceptsforrothebasisof mathematicalfotmalismof thetheoryof
multifractals.
3.THEORY OF ELASTICITY OF FRACTALS
In themajorityof investigationsof elasticityof fractalstheelasticbehaviorof self-similar
DimensionDq
r(q)
dr
f(a)
=
q
.a +r(q)
q
a=--
dq
Fractaldimension
O
DF
=
Do
=
dimHM
dimHM ao
fmax=dimHM
Iníormationdimension
I(r)
1
DI
=
Dl
O
al
=--

=
al
=
1
*
Inr
CorreIationdimension
-De - (dDq)
2
De
=
D2
-De
2a2- De
dq q=2
+00
Uperlimit Doo
InP+**
f-+O'"-q.amin
-+amin=--
Inr
Lowerlimit D-oo
InP- **
f-+O
-00
'"-q.amax
-+amax=- -
Inr
512
A.S.BALANKINANDP.TAMAYO
TABLE II.Comparisonoí Poisson'sratio v,calculatedusinganalyticalrelationsbip,witb tbe
computervalues,baBeaontwo-dimentionalelasticra.ndomnetworksandwitbexperimentalvalues
oí aerogelSiO2.
structuresisstudiedbycomputersimulationmethods[19-22].It is usualtoconsidertwo
limitingcasesof isotropica~dcentraltoreesgoverningelasticity.It is shownin Ref.[22]
that problemsof elasticityof statisticallyself-similarnetworksconsideredin theselimits
belongto twodifferentclassesof universality.Moreover,numericalexperiments[19,20]
showthat elasticrandompercolationnetworkson aplane (d
=2),
whosedimensionality
is L < 0.2~c(~cis thecorrelationradius),havecharacteristicnegativevaluesof the
Poissonratio,1),whereasfor networksof sizeL >0.2~cthisratioispositive.Thelimiting
valuesof 1)(L/~c)areuniversalin thelimitsL/~c -+OandL/~c -+00(seeTableII),
i.e.,theyareindependentof theratioof thelocalelasticityparametersusedin numerical
calculationsof theelasticpropertiesof randompercolationnetworks.
Thesecondapproachtosimulationof elasticityof multifractalsis theclassicalconcept
of theentropicnatureof theelasticbehaviorof multifractalpolymernetworks[41,42].
Finally,therehavebeenfrequentreferencesin theliteratureof theanalogybetween
elasticbehaviorof multifractalpolymernetworksandrandomsprings,whosesizeexceed
certaincharacteristicscaleof lengthLF [43,44].This analogyhasbeenusedby Web-
man[15]to propasea heuristicpictureof elasticdeformationof fractals,accordingto
whichapplicationof forceF causesdeformationof fractalonlyayerdistancesexceeding
thecharacteristicscalinglengthLF,whichdependsontheforceF.Therefore,anexternal
forceF actingon an elasticallydeformedfractalcreatesa newcharacteristicscaling
length[15].Elsewhere[29-31]weusedthis pictureto developthermodynamicaltheory
Two-dimentionalrandomnetworkoí size
L x L neartbepercolationtbresbold
Properties
(c - correlationlengtb) AerogelSiO2
L/c -+00 L/c -+O
Elasticityoí network
Bonds,determining
Connectednessoí
is determinedby elasticityoí network
randomnetwork
danglingbonds aremultiduplicated
Fractalcluster
DF,measuredby
smallangleneutron
Dimensionoí Randomwalk
scatteringand
Fractaldimensionoí
geodesicline
dimension'Dw=0.67
molecularadsorption
elasticbackboneDF
Dr =1.1:1:0.02[22] [22] 2.3:1:0.1[44]
v
=
DF/(d -1) -1
0.1:1:0.01 -0.33 0.15:1:0.05
Poisson'sratio
(resultsoí numerical
simulationand
experimentaldata)
0.08:1:0.04[20] -1/3 [20] 0.12:1:0.08[44]
FRACTAL SOLID MECHANICS
513
of elasticityof multifractalsandexplaincertainfeaturesof the elasticdeformationof
self-similarstructures.Belowweconsiderthistheory.
It iswellknownthatclassicaltheoryofelasticityofsolidisbasedontwoexperimentally
establishedfacts[43]:
1) Hooke'slaw,accordingto whichrelativestrainfll isproportionalto theactingstress
0"11:
fll
=
O"U/E.
2) Poisson'slaw,postulatingthe effectof transversestrains
i
f.L
=
-vfll
(0";1
=
O).
In developingthetheoryofelasticityoffractalswehavealgostartedwithtwopostulates:
1) WhenanelasticallyisotropicmultifractalisdeformedbyanexternalforceF,aunique
newcharacteristicscalinglengthLF appearsandtheforceobeysthefollowingrela-
tion:
F= BU -TBS,
BLF LF
(5)
whereU is theinternalenergyandS is theentropy.
2) In thereversibledeformationcase,an elasticallyisotropicmultiffactalretainsself-
similarity,i.e.,thelawdescribingthechangein densityp asa resultof elasticdefor-
mationis similarto\pelawdescribingthechangein p dueto geometricchangein
thedimensionalitiesof a multifractalstructure:
p(F1)
=
(
BLF1
)
-O<- -O<
p(F2)
...
BLF2 -)..F
'
a=
d
-
DF,(6)
whereDF is themetricdimensionof structure.
In the caseof uniaxialdeformationit followsfromEq.(6) that the changein the
dimensionalityof a fractalalongthedirectionof theforceF,whichis )..X
=
Lx/lx,is
accompaniedbya changein its transversedimensionalityin orthogonaldirectionsof the
surroundingd-space)..i
=
Li!li,wherei
=x,y,....Thesearerelatedto)..X
=
)..Fby the
followingrelations:
)..i
=
)...L
=
X;VF
=)..¡/F,
i = 2,3,...,d,
(7)
wherea =1
-
(d- 1)VF,sothat
In)...L- DF - 1.
VF= -In )..F- d - 1
(8)
514
A.S.BALANKINANDP.TAMAYO
Equations(7) and(8) aresatisfiedin thegeneralcaseof n-dimensionaldeformationof a
multifractalstructurein a d-space.For example,in thecaseof biaxialdeformationof a
fractal(1:::;DF:::;3) in three-dimensionalspace(d
=
3)it followsfromEq.(6)that
\\-VF
.l\z=.I\F,
AF=(
A A
)
1/(I-VF)
x y,
(9)
where1/FandDF arestill relatedbyEq.(8),whichis satisfiedalgoin thecaseof triaxial
deformationof a fractalin three-dimensionalspacewhen
AF
=
(AIA2Aa)l/a,
a
=
d
-
DF
=
1- 21/F,
(10)
andthelaw(6)isvalidjhere,pexX'FOI.Note,thatthecoefficientoftransversedeformations
of multifractals1/F
=
InA1-/InAF isequaltothePoisson'sratio1/
=-[(A~-l)/(AJ -1)]1/2
only in the limit of infinitely small strains 
=
IA2
-
111/2~ 1.
Therefore,it followsthat whentheconditions(5) and (6) arevalid,thetransverse
deformationfactorof anelasticallyisotropicmultifractalisdefineduniquelybyitsmetric
dimensionof Eq.(8).It isclearfromTable11,thatthevalueof 1/calculatedfromEq.(8)
agreewellwiththeresultsobtainedin Refs.(19]and[20]by numericalmodelingof the
elasticitybypercolationnetworksonaplane(d
=
2) andwiththeexperimentalvalueof 1/
foraerogelSiO2reportedinRef.
[44]
andobtainedbystudyof propagationof longitudinal
andtransverseelasticwaves.
It followsfromEq.(9) that theconditionof incompressibility(a ==O)is satisfiedfor
m~ltifractals,whosemetricdimensionis thesameas thedimensionof thesurrounding
euclideanspace(DF
=
d),fromwhichit followsthat
1
1/max
=
d- 1.
(11)
,.
We caneasilyseethat if d
=
2 and3,theequality(12)is identicalto theconditions
of incompressibilityof two- (1/max
=
1) and three-dimensional(1/max
=
0.5)elastically
isotropicsolidsderivedin theclassicaltheoryof elasticity(seeRef.[43]).
Usinglinearrelationshipbetweenentropyandinformation
[45],
andthedefinitionof
theinformationdimensionDI of multifractals,wecanrepresentthechangeintheentropy
asaresultof elasticdeformationof multifractalin d-dimensionalspacewithlaw(6)valid,
bythefollowingexpression:
d
~S=-BLAfI_d,
i=1
(12)
wheretheparameterB is independentof Ai andcanbedetermined,for example,using
theapproachof Ref.[46].
Accordingto thedefinitionof correlationdimensionDe,thechangein theinternal
energyU(Ai) of multifractalduringthereversibledeformationcanbewrittenasfollows:
~U
=
-C(A~C- 1),
ae
=
d- De,
(13)
FRACTAL SOLID MECHANICS
515
whereC is constant.Substituting(8) and(12),(13)into (5) andusingthermodynamic
equation[30]
DI(d
-
DF)
=
Dc(d
-
De)
withtheconditionF
=
Ofor'xi
=
'xF
=
1,wehave
F
=C{DI'xfr-l
-
DI[DF- (d- 1)],X~Dr[DF-(d-l)]-1- Dc(d- Dc)'xf-Dc-l}.
(14)
In thecaseof regularfractals,wehave
DF
=
DI
=
De
=DH,
(14)
whereDHisHausdorff-Besicovitchdimension.Thenit followsfromEqs.(8)and(14)that
F
=C{'xfF-l - [DF- (d-1)],X¡vDF-l- (d- DF)'xf-l}.
(16)
Thetruestress(TIis relatedtoF by
(Tn
=
F,X~-a
=C{,XiDF-l- [DF- (d-1)],XfF(1+VF)-d- (d- DF)}'
(17)
whichallowsfor thechangein the afeaof the (d- 1)-dimensionalcrosssectionof a
deformedfractalin aplane(hyperplane)orthogonalto thedirectionof forceF.
Similarly,usingEqs.(5)-(15),wecanobtaintherelation(Tii('xjj)for n-axial deforma-
tion of an'n-dimensionalfractalin thed space.We caneasilyseethat in thelimit of
infinitesimallysmallstrainskil
=
1'x1-111/2~ 1theserelationshipscanberepresented
bytheclassicalformof generalizedHooke'slawfor d-dimensionalsolid:
,.
(
d-l
)
oo=.!..(T"-V~(Too
n
E
n
~
JJ
.
j=1
(18)
This allowsusto findtherelationshipsbetweentheelasticmoduli(i.e.,Young'smod-
ulusE,shearmodulusG,bulkmodulusB,longitudinalelasticmodulusCn) of fractals,
whicharederivedbyanalogytothederivationofcorrespondingrelationshipsinthetheory
of theelasticityof solids:
E
d-1 B=~,
G= 2DF'd(d- DF)
1+(d- 2)(d- DF).
Cn
=
2G (d-1)(d - DF)
(19)
(20)
SubstitutingEq.(8) in thesystem(19),(20)for d
=
2 and3 wecanseethat these
expressionsareidenticaltothosefortwo-andthree-dimensionalelasticallyisotropicsolids
(seeRef.[43]).
516
A.S.BALANKINANDP.TAMAYO
Noticethat,incontrasttothetransversestrainsinsolíds,whichaccordingtoPoisson's
lawoccurwhen(]'.l
=
O,thetransversedeformationoí íractals,whichensurethatthe
law(6)issatisfied,occurundertheactionoí stressesthatappearduetothefinestructure
oí a íractal,whichcanbedemonstratedin a clearmannerbyconsideringtheprocessoí
elasticdeíormationoí theKochcurves(DT
=
1,1
<
DF
<2,d=2).
4.FRACTAL THEORY OF THE RUBBER ELASTICITY OF POLYMERS
Experimentaldataon thereversibledeíormationoí elastomersareusuallyinterpreted
in termsoí classicaltheoryoí rubberelasticity(see,íor exampleReís.[41,42]).Classical
theoryoí rubberelasticitywasdevelopedinthe40'sindependentIybyanumberoí workers
(seeReís.42]).The mainsimplificationoí theclassicaltheoryis the assumptionthat
thesubchainsoí a polymerchain(i.e.chainsbetweentwoneighboringcrosslinks)can
takeon with equalprobabilityan arbitraryconíormationcomparablewith the given
distancebetweentheendsoí thesubchain,i.e.,betweenthecrosslinksthat límit the
subchain.The elasticityoí polymerschainis in thiscaseoí purelyentropicnature.It is
easyto understand,consideringthatthenetworkis deíormed,thatthedistancebetween
crosslinksarecorrespondinglyalteredthusdecreasingthesetoí possibleconíormationsíor
theaggregateoí subchains.Nowit isobvious,thatthesuccessoí classicaltheorydoesnot,
in itselí,implythat thechainsoí polymernetworksobeyGaussianstatistics,whichlíes
at thebasisoí thistypicalmean-fieldtheory.It iswellknown[46] that Gaussianstatistics
is characteristiconlyíor networkspreparedby thecoalescenceoí concentratedsolution
oí chains when they are compressedor weakly (A
<
1.2) extended.Such structures are
characterizedby iníormation dimensionDI
=
2 and the generalrelationshipoí Eq.(12)
reducesto theclassicalíorm
,.
(
d
)
~S
=-B
¡:::A;
-
d
,
~=l
so that ií the 11= 0.5 (thesecondassumptionoí classicaltheoryoí rubber elastic-
ity [41,42]),wehavea classicalexpressioníor Fl(Al) intheíorm
Fl
=~
(Al - A1"2),
(21)
whereE is theYoungmodulusproportionalto thetemperatureíor whichtheexpression
in termsoí theparametersoí polymerstructurehasbeensubjectedto repeatedrefine-
ments[42,46].Relation(21)agreeswellenoughwithexperimentaldatainthecompression
region (A
<
1),butat A
>
1considerabledeviationsappearright away
[42].
If 1
<
Al
<2,
the graph oí relation (21) usually líes aboveand íor Al
>
2 well belowthe experimental
curveF(Al),whichin therangeAl >4 is usuallydescribedby theasymptoteFl exA~
(seeFig.1).
Traditionally,therefinementoí therelation(21)is madebyphenomenologicalmodifi-
cationsoí theentropictheory,themainprogressin whichis summarizedin Reís.[42,47].
FRACTAL SOLID MECHANICS
517
6
1
2
5
4
/1
1
1
I
1
-r-
1
1
1
1
1
I
1
/
1
-+-
1
I
1
.
1
1
//
/
/",
,
l/
.-/
~./l 3
éi
D.
~ 3
-
/
./
u.
2
,
5
A
FIGURE
1.FunctionF('x¡) for uniaxialstressoí rubber:1) CalculationusingEq.(25)for E
=
0.36MPa (v
=
0.5);2) calculationusingEq.(24)for E
=
0.34MPa andv
=
0.48;3) calculation
usingclassicalformula(21)for E
=
0.4MPa (v ==0.5).(Pointsrepresentexperimentaldatafrom
Ref.[48]).
/
.///
04
1
,.
2
3
4 6 7 8 9
The requiredprecisionin matchingthecalculationsto experimentsis attainedby the
useof additionaletchingparameters,whichareessentiallyof thefitting type [42].In
this approachof Ref.[48],whichis basedontheuseof empiricalrelationof theelastic
potentialon theinvariantsof strainandtemperature,whichensuresanydesiredpreci-
sionof approximationto theexperimentaldataif sufficientnumberof thefitting-type
matchingparametersis selected.Apart fromthe largenumber,and not alwaysclear
physicalmeaningof thematchingparameters[41,42,47],themainshortcomingof both
suchphenomenologicalmodificationsof theentropictheoryandtheempiricalmodelsof
theelasticpotentialistheneedtousedifferentvaluesof thesameelasticparameterssuch
as Youngmodulusto describetheexperimentaldataobtainedunderdifferentloading
518
A.S.BALANKINANDP.TAMAYO
conditionsandalgoto describethesamedata,but withinthe frameworksof different
modificationsof theentropictheoryor of theelasticpotential[47].This is dueto the
indeterminacyof theabsolutevaluesof theelasticparametersandof therelationships
betweenthem,whichdonotsatisfytheexpressionsof Eqs.(19),(20)evenin thelimit of
infinitesimallysmallstrains.
Belowweconsiderproblemsof rubberelasticityof theelastomers.Elastomers,in par-
ticularpolymers,areknownto haverandomseU-similarmultifractalstructure[11-15].It
is natural,therefore,to describerubberelasticityby usingtheresultsof thetheoryof
elasticityof multifractalsdiscussedabove.
Sincein thecaseof seU-similarstructureswealwayshaveDe ~ DI,it followsfrom
Eqs.(12)and(13)that theseshortcomingsof theentropictheoryof rubberelasticityof
polymersareduetotheconflictbetweenthetwomainassumptionsoftheclassicaltheory:
1) the Gaussianstatisticsof polymerchains,whichis validin the caseof structures
whoseinformationdimensionis DI
=
2;
2) theincompressibilityof elastomers,whichaccordingto Eq.(6) is satisfiedwhenDF
=
d
=
3
=DI.
Theformerassumptionisvalidinthecaseof longpolymerchainsandnetworksformedby
crosslinkingofconcentratedsolutionoí chains(DI = 1,DF= 2,md= 3)
(seeRefs.[14]
and[15])forwhichthePoissonratiois-accordingtoEq.(8)- zero(v
=
O),whichimplies
constancyof thetransversedimensionalitiesin thesurroundingspacewhena chainis
subjectedto uniaxialdeformation.This canbeeasilydescribedbyconsideringelongation
of sfronglytwistednondeformablefilament(dr
=1,
DH
=
2,d
=
3).SubstitutingDF
=
2
andv
=
OintoEq.(16),wehave
F
=
E(A1- 1),
(22)
"
whichis identicalwiththeclassicalresultfor a longpolymerchain[41,42].
Networkswhichswellin a goodsolventarestronglynon-Gaussian[46],so that the
dimensionof thestructureoí realpolymersformedbyinterpenetratingblobs[15,41,42]is
withintherange2<DH <3,andin generalwehave
1~Dq~Dq-1~D2
=
De ~DI
=
DI
~Do
=
DF
~3.(23)
Assumingin thefirst approximationthat De
=
DI
=
DF andsubstitutingEq.(8) into
Eq.(16),weobtaintherelationshipbetweenthenominalstressF andthestrainfactor
Al in thecaseoí uniaxialtension(compression)of anelastomer:
E
F
=
{
A1+2v
2
\-1-2v(1+v)
1+6v+4v3 1 - vl\l -(1-2v)A¡2V}.
(24)
Thisexpressiondiffersfromtheclassicalexpression(21)eveninthelimitof incompressibly
deformedmaterial,whenEq.(24)reducesto
F
=
~
(
A2- A-2,5
)
4.5 1 1,
(25)
FRACTAL SOLID MECHANICS
519
obeyingtheasymptoticexpressionF exA~,whenAl
»
1,whichagreeswellwithexperi-
ments.
It is clearfromthegraphsin Fig.1that thecalculationbasedonEq.(24)is in good
agreementwiththeexperimentaldatarightupto Al
=7,whencracksappearinthede-
formedmaterialandtheeffectivevalueof thetransversestrainfactorof Eq.(24)becomes
largerthen0.5.
5.FRACTAL FRACTURE MECHANICS
A solid bodyrespondsto extremeloadingby undergoinglargedeformationandforfrac-
ture.This phenomenoni.e.,lossof contactbetweenpartsof thebody,is a problemthat
ourcivilizationhasfacedfor aslongastherehavebeenmanmadestructures.Theprob-
lemactuallyis worsetodaythanin thepreviouscenturies,becausemorecangowrong
in our complextechnologicalsociety.At thesametime,the natureof processeswhich
determinecrackformationandgrowthin materialsis far frombeingunderstood.The
failureof materialsunderstressis acomplexprocessinvolvingabroadrangeof physical,
chemicalandsometimesbiologicalprocesses.It wasfoundthat fracturesurfacesareof
fractalcharacterin awiderangeof spatialscales[11,23-25].Moreover,it wasshownthat
thefractaldimensioncanberegardedasa measureof fracturetoughnessin solids[25].
Now,quantitativeanalysisof fracturesurfaceshasbecomeanimportantfeaturein the
processofobtainingbetterknowledgeofthemicrostructuralprocessesinvolvedduringthe
deformationandruptureof materials.Somesimplemodelsof fractalgeometryof failure
andcrackgrowthwereproposedin worksof P.Meakingetal.
[26],
H.Herrmann[27],M.
López-Sanchoetal.[37],A.Balankin[50],andotherauthors.
Mostrealfailureprocessesof practicalimportanceexhibitrichphenomenologyextend-
ingovera widerangeof lengthscalesfromtheatomiclevelto theoverallsizeof the
sampleor structure.Becausemostfailureprocessesinvolvecomplexinteractionbetween
largenumberof processes,it hasin mostcasesbeendifficulttodevelopunderstandingon
fundamentallevel.Nevertheless,considerableadvanceshavebeenmadetowardsdevelop-
ingsatisfactoryunderstandingof mechanicalfailureprocessesonphenomenologicaland
statisticalbasis.Evenin idealhomogeneousmaterials,a complexnonlocalstress-strain
fielddevelopsas thematerialbeginsto fail andan understandingof the evolutionof
thestress-strainfieldis an importantingredientin developingbetterunderstandingof
materialfailure[4].The self-similarcharacterof evolutionof defectsat variousscales
allowsto usenewmethodsof theanalysisin termsof fractaltheory.
Thecausesandlawsof formationofthefractalgeometryoffractureofinitiallyhomoge-
neoussolidshavebeenstudiedbyA.Balankin[31-35].Heproposedaquantum-statistical
approachto solidmechanicsin whichfractalnatureof fracturefollowsfromfundamental
principIesasa resultof thecollectiveexcitationsof theatomsin crystallattice[4].It was
shownthat fromthetopologicalstandpoint,thedifferencebetweenductileandbrittle
fractureis that in brittlefracture,thecrackfront (crackhasa smoothsurface),whose
metricdimensionliesin therangefrom1 to 2,is fractal (at stresses(l > (lC classical
Griffith'scrackwitha smoothfrontandsurfacepropagates),andin ductilefracturethe
surfaceof crackswhosemetricdimensionlíesin therangefrom2 to 3 is fractal.This
520
A.S.BALANKINANDP.TAMAYO
differenceisattributabletothedifferencein thekineticsof self-organizationof dissipative
structurescomingfromtheinherentpresenceinmaterialsof defectsof differentdimension
suchaspointdefects,dislocations,disclinations,etc.Someconclusionsconcerninginter-
pretationsof solutionto theproblemsof lineartheoryof nearcrackedgewereobtained
on thebasisof fractalconceptin theworksof A.Balankinet al.(seeReí.[4D.It was
shownthat thephenomenonof fractaltopologyof fractureof solidsis a consequenceof
Poisson'seffectof transversestrains.Someneweffectsin thephenomenaof fracturehave
beenpredictedbyusingmethodsof fractalmathematics.
Nowit is obviousthat fractalconceptsandmultifractalanalysisareveryusefuI in
understandingnatureof thefracturephenomena.However,since,obviously,it is still toa
earlyto speakaboutthestructureof fractalfracturemechanics,it seemspreferableto
developpartialtheories,describingthebehaviorof certainclassesof phenomenabyusing
fractalconcepts.Belowweconsiderthefoundationsof thefractalapproachto fracture
mechanics.The mostimportantexperimentalandtheoreticalresultsof fractureanalysis
whichwereobtainedbyusingfractalmathematicsarepresented.
5.1Fractaldynamicsoffractureof modelelasticlattices
Variousfracturemodelshavebeensuggestedandin thesetheedge,thesurface,or the
distributionof crackswerefoundto beself-similar[25-29,35-39].A fundamentalrelation
betweenfractalgeometryantheself-similarityof thefractureprocesseswaspointedout
in Refs.[31,34].
.Fractureofplanarlatticeswasconsiderinworks[15,19,20,50].Theirdynamicshasbeen
describedbyequationsof elasticityof continuousmedia
(.1+'1')8;(~8jl'j) +l'(~aJ ) 1"~O,
(26)
whereJ.tiarethecomponentsof thedisplacementfieldand8i is thepartial derivative
withrespecttothei-th componentof thepositionvectorr.Thedescriptionof thecrack
propagationafterbreakingthefirstbondyieldstheruleaccordingtowhichtheprobability
of breakingof thebondbetweenadjacentsitesi andj isproportionalto them-th power
oí thestressactingonit,
m
Pij exO'ij'
(27)
It wasestablishedin Refs.[15,19,20,50]that thefractaldimensionof self-similarconfigu-
rationsof cracksis independentof thenatureof loadingwhichis modeledbyspecifying
therelevantboundaryconditions,andis governedby thevaluesof Poisson'sratio (see
Fig.2a)andprobabilityexponentm(seeFig.2bandTable111).
Wecanseefromthegraphin Fig.2athat thedependenceof thefractaldimensionof
cracksDF(1I)onPoisson'sratio11is linearform
=
1andit isgovernedbythedimension
of thefieldof inhomogeneousstrainsDF
=
(d-1 )(1+11)
=
1+
11
(seeRefs.[4,31D.Wecan
easilyseethat thelast relationshipis identicalwithEq.(8),describingtherelationship
0.4 0.6 0.8
0.2
v
FRACTAL SOLID MECHANICS
521
2 te"
\
""""
1.8
1.6
,
,
Dr:
1.4
--',2
-----
1.2
b
1
O
0.5 1.5
m
FIGURE 2.
Fractaldimensionoí crackconfigurationsasaíunctionoí parametersoí thestochastic
model (26),(27) íor íractureoíelastic lattice:a) DF(V) calculatedassumingthat m
=
0.5 (curve1),
m=1(curve2),m=2 (curve3);b) DF(m) calculatedassumingthat v
=
0.2(curve1),v=2/3
(curve2),andv =0.9(curve3).(ContinuouscurvesarecalculatedusingfirstEq.(31),andthe
pointsaretheresultsoí computersimulationsusingEqs.(26),(27)íromReís.[23,24](.) and[32]
(.).)
betweenmetricdimensionandcoefficientof transversedeformationsof themultifractals.
This resultis a reflectionof thefeasibilityor representingtheequationsof thelinear
theoryof elasticityat smallvaluesof ij in theformof Eq.(5) and(6),wheretheroleof
LF is playedbythecharacteristicinhomogeneityscaleof thestrainfield.
In viewof shearrigidityof solids,wecanexpectheterogeneousfluctuationsof density
(bothspontaneous-quantum andthermal- andthoseinducedby an externalagent,
bya changein boundaryconditions,andbybondbreaking,asin themodeldescribedby
Eqs.(26)and(27)).tobealwaysaccompaniedbytheappearanceof shearstressesjsince
themínimumscaleof stableinhomogeneousfluctuationsof densityconsiderablyexceeds
theinteratomicspacing,thespatialdistributionof heterogeneousfluctuationsof density
andsheardeformationsin adeformedsalidis scale-invariant.Therefore,thedistribution
andtheconfigurationsof cracksin adiabaticfractureshouldhaveseU-similarstructure.
This is confirmedbytheresultsof numerousinvestigations[19-20,50]andis illustrated
bythedatashownin Fig.2.
It shouldbenotedthatmodelingofcriticalfluctuationsinthecaseof aone-dimensional
harmonicor anharmoniccrystal[51],whichischaracterizedbyv
=
Obecausethereis no
effectivetransversestrain,yieldsasmooth(DH
=
d-1
=
1) fracturesurface,whereasthe
modelof noninteractinganharmonicoscillatorsplacedin a thermostat[52] corresponds
to theoppositelimit v
=
VM
=
1/(d- 1),whichensuresconstancyof thevolumeof a
deformedmedium,leadingto a homogeneity(DHexd) of thedistributionof microcracks
ayerthevolumeof thesample.
a
I
1.8
1.6
D
1.4
1.1
1
o
522
A.S.BALANKINANDP.TAMAYO
TABLEIn.Comparisonof fractaldimensionsof crackscalculatedusinganalyticalequationdi
=
1+
11mwiththosecomputerbasedontwo-dimentionalnetwork(d
=
2)withA
=
JL (11
=
2/3).
In the dilationalmodelof fracture[53]
thefluctuationvolume'i~,whichoccursin
Zhurkov'sexpressionfor thelifetime-under-loadT,is governedby thecriticaldimension
of dilationid.Obviously,idis thenaturallowerlimit Lo of theself-similarityof fracture
surface.Comparisonof anestímateof id [53] with thevaluesof Lo obtainedin fracto-
graphicinvestigations[23,49] showsthat,with satisfactorydegreeof precision,wecan
assumethat Lo =id.
In thecourseof fracturetheeffectivevalueof thetransversestrainfactorlIeffchanges
becauseof theaccumulationof damageandbecauseof relaxationof stresseson rough
crackfacesand duringplasticdeformation.This is the reasonfor the formulationof
multifractalself-similarcrackconfigurationandtheir surfaces,andfor thedistribution
of theirsizes.The fractaldimensionof themajocrack(dependingonthemechanismof
fracture,wecanspeakeitherof thesizeof thesurfaceor of theconfiguration)isgoverned
bytheeffectivevaluelIeffandin thecaseof fractureof three-dimensionalsolidsit isgiven
Effectivefractaldimensionsobtainedfromthecrack
growthmodels
Typicalcrack
Thevalueof
(brokenbonds)
m-parameterof generatedby
bond-breaking
meansof the
Resultsof numericalsimulationfor
probability
stochastic
differentboundaryconditions
exponentp¡
,...,
Ei
fracturemodel
(11
=
2/3) [23]
Dilatational
Analyticalvalue
strainEl ShearstrainES
1+11m
Dispersion
fractureof elastic
«1
lattice 2.00 2.00 2.00
Dispersion
fractureof elastic
0.5 lattice 1.9:1::0.1 1.9:1::0.01 1.82
Multiplefracture
1.0
of elasticlattice
1.66:1::0.05 1.65:1::0.05
1.67
Growthof fractal
2.0
crack
1.45:1::0.05
1.40:1::0.05 1.44
Propagationoí
00 thesmootJtcrack
1.00
1.00 1.00
FRACTAL SOLID MECHANICS
523
by
DF
=
2(1+Veff),
(28)
whereVeffis a functionalof thefractureprocess[54].
Turningbackto the stochasticmodelof fracturedescribedby Eqs.(26) and (27),
weshall considertheprocessof formationof cracksin thepresenceof uniaxialstrain
(1:11
=
const,1:.1
=
O) in a modelof elasticlatticeundertheactionof a stress0"11.We
shall assumethat propagationof crackcanbedescribedas a sequenceof mappingsof
thestateof latticein thephasespace[4,30].Sincein thestochasticmodelof fracturethe
cracksareknownto haveanirregularroughsurface(configuration),it followsthat the
mapping,describingpropagationof crack,isofthecompressivetype.This happensdueto
thedissipationof energyof theelasticstrainsonaroughsurfaceof thecrack.This energy
dissipationby relaxationof stressesat brokenbondshasbeenestablishedin themodel
describedbyEqs.(26)and(27)bynumericalmodeling[15,19,20].Theoreticalrelaxation
ofstressesonarough(althoughnotseU-similar)cracksurfacehasbeendiscussedindetail
in Ref.[55].
It wasshown(seeRef.[43])that,in thecourseof crackpropagationunderseU-similar
conditions,thefractionof energyof theelasticstrainsstoredbya solidandlostin relax-
ationof stressesonseU-similarcracksurface,is

=
1-".-(3
,
(3
=
DF
- (d-1),(29)
i.e.,
DF=(d- 1)- ln(l- r¡)
In".
,
(30)
where".
=
Li+1/Li is theself-sim'ilarityparameterrepresentingthehierarchyof thespatial
scalesLi of thestructurallevelsof thefractureprocess[4-6].
In thecaseof uniaxialdeformationofanelasticlatticewecanexpecttransversestresses
O"i
= VO"II,
wherei
=1,2,...,(d- 1).It followsfromthedistribution(27)that,when
a crackpropagat~s,breakingof bondsundertheactionof stresses(O"ij}.Loccursonthe
average(d- l)vmtimeslessthanundertheactionof stresses (O"ij)lI.Here,thesymbol
(...) denotesaveragingoverthestatesof thelattice.Averagingisessentialbecauseof the
redistributionof thestressesattheremainingunbrokenbondstakenplaceinthecourseof
crackpropagation.Usingtheexpressionforthefractaldimensionof aphasepathinterms
of thecharacteristicnumbersA+andA- of compressivemappingbetweenthestatesof
thelatticein the"coordinate+energy"space,andbearingin mindthat
A+A- <X(1- 1:11)(1+Vml:lI)d-l <X(1- 1:11)'\
wherea
=
1- (d-l)vm,weobtainthefollowingexpressionforthefractaldimensionof
seU-similarconfiguration(d
=
2) or forthesurface(d
=
3) of crackformedin accordance
withthemodeldescribedbyEqs.(26)and(27):
DF
=
1+vm,DF=2(1+vm).(31)
524
A.S.BALANKINANDP.TAMAYO
Theseresultsarevalidfor d
=
2and3,respectively.It isclearfromthedatapresentedin
Fig.2 andin TablenI that thecalculationcarriedout onthebasisof (31)is in a good
agreementwiththeresultsobtainedby numericalmodelingof thefracturedynamics.
The propagationof fractalGriffithcrackwasconsideredin Refs.[31,54].It wasshown
thatnearthetipoffractalGriffithcrack,stressesaredescribedbythefollowingasymptotic
expression:
q ex:R-a
,
d-DF
a=2
(32)
whereR is thedistancefromthecracktip andDF is fractaldimensionof a self-similar
cracksurfaceof Eq.(32)or thedimensionof thefractallineatthefrontof asmoothcrack
equalto DF
=
(d
-
1)(1+Vefr)
- 1.
Wehaveconsideredself-similarGriffithcrackin themodeldescribedbyEqs.(26)and
(27)in ardertodetermineVeffandDF of Eq.(28).In thed
=
2casethenumberof bonds
in a crackof sizeF obeysM ex:R2,whereasthenumberof bondsin a crackof sizeR is
proportional,accordingto Eq.(27),to K = MP(R),where,bearingin mindEq.(27),
wehaveP ex:qmex:R-ma,i.e.
K ex:R2-ma.
(33)
Ontheotherhand,bydefinitionwehaveK'"RDF,sothatcomparisonofEqs.(32),(33),
and(31),showsthat in themodelof Eqs.(26),(27)fractalGriffithcrackpropagatesif
m=2,t.e.,
Veff=v2.
(34)
This allowsustoexpress\!1edependenceof thestressintensityfactorK¡ onR obtained
in Eq.(32)for d
=
3in theform
K¡ ex:qR(1-2v2)/2
,
K¡ ex:qR(1-v2)/2.
(35)
The first of therelationshipsin Eqs.(35)correspondsto thepropagationof crackwith
self-similarityandthesecondcorrespondsto thepropagationof smoothcrack,thefront
of whichis fractalcurve.Notice,that therelations(35)agreewell with theresultsof
experimentalinvestigationsreportedin Ref.[25].
5.2F'racture01materialswithmultilractalmicrostructure
The failureof materialsandstructuresunderappliedloadis a subjectof bothpractical
importanceandconceptualdifficulty.Microscopicfailureplayfundamentalrolein many
systemsof industrialimportancerangingfromaircraftstructuresandpressurizednuclear
reactorsto cracksin undergroundGil reservoirsandin ceramicsand fiber composites.
Analysisof fractureof stronglyheterogeneousmaterials(aerogels,composite,quasialloys,
amorphousandporousmaterials,rocks,geophysicalmedia,etc.)is usuallycomplicated
by manyfactors:heterogeneityandanisotropyof mechanicalproperties,redistribution
FRACTAL SaLID MECHANICS
525
of stresseslinkedto thegenerationandgrowthof macroscopicdiscontinuities,etc.All
thesefactorshavedifferenteffectonfractureprocessesandcorrectdescriptionof failure
transitionfrommicroscopicto macroscopiclevelis possibleonlyonthebasisof adequate
procedure,algotakingintotheaccountthestochasticnatureof mechanicalpropertiesof
a real material.Usual treatmentsbasedon continuumelasticitytheorydo not provide
simpletoolsfor discussingessentialnonlinearitiesof thisproblem.
Manyphysicalsystemsarebeingmanufacturewhichhavefractalor multifractalstruc-
turein thewiderangeof spacescales.Threeexamplesof physicalsystemsforwhichthis
discussionis of interestare:1) randomcompositematerial,2) colloidalaggregatessuch
asgoldandsilicaaggregates,and3) geophysicalmedia.Thesestructuresaremadeupto
unitsthatareconsiderablylargerthanatomicsizeandremainstableintheconfigurations
in whichtheyareprepared.Othermaterialsto whichthetheorydiscussedbelowcould
beappliedaremicroscopicallydisorderednetworkmaterialssuchasgels,polymers,and
glasses.The majorityof themcanbe regardedas an ensembleof fractalsof different
dimensionscharacterizedbydifferentweights.
Whena solidbodyfractures,crackpropagationoverdistanceD..Lis ensuredby the
releaseofelasticenergyU(D..L),whichisspentontheformationof fracturesurface.When
smooth(dT
=
2) crackpropagatesin three-dimensional(dT
=
3) linear-elasticmedium,
theasymptoticformsof thedistributionof stressesO'ij(r),anddisplacementsof crack
edgesui(r),asa functionsof distancer to its movingtip aregivenby [10]

O'ij
=
...¡rcp(()),
- K¡V4 f(()),
Ui
-
E
(36)
whereK¡ is thestressintensityfactor.Thespecificenergy,therefore,is
,.
U(D..L)
=
consto
Uo
=
D..L
A fundamentallydifferentsituationarisesduringfractureof materialswithmultifractal
microstructure.Wedescribecrackpropagationin multifractalstructurewithintheframe-
workof nonlinearfracturemechanics,utilizingJ andr invariantintegrals(seeRef.[10]).
After almostliteral repetitionof theoperationsof Ref.[10]weobtainthe asymptotic
forms
K
_/)<
O'ij rv Fr
,
KF r/3,
Ui
rv
E
(37)
where
a
=
ndF
-
DF
n+1'
(J
-
1+n(DF+1- dF)
- n+1'
(38)
andn is theexponentof unit strainsij
rv
[l/n asa functionof stress
[ ]
1/2
[= (O'ij- 1/20'kkÓij(O'ij- 1/30'kkÓij).
(39)
526
A.S.
BALANKIN ANDP.TAMAYO
It iseasyto seethat in thecasen
=
1,in thelimit
dF
=
d
=3,
DF=dT=2
theasymptoticforms(37),(38),and(36)areidentical.In thecasedF
=
d
=3,
DF
=
dT
=2,
butforn
#
1,from(37)and(38),weobtainthewellknownresultof Ret [10]
for a smoothcrackin nonlinearelasticmedium.Whenn
=
1anddF
=
3,butDF >2,
Eqs.(37)and(38)becomeasymptoticexpressionsobtainedin Ref.[56]forfractalcrackin
linearelasticsolidoWenotethat (37)and(38)suggestthatbrittle(quasibrittle)fracture
of materialswithmultifractalstructureispossibleonlyif themetricdimensionof fracture
surfaceDF is lessor equalto themetricdimensionof multifractalstructuredF.
Usingprobabilityfracturecriteria(27),andliterallyrepeatingtheoperationsdescribed
above,weobtain
DF
=
2(1+vm),
(40)
whichcoincideswiththeexpression(28)forthefractaldimensionof thesurfaceof acrack
in solidoIn particular,m
=
2,correspondsto Griffithfractalcrack,withinthelimitsof
self-similarityLo
<
AL
< L M.Therefore,the specificdensity of elastic energyreleased
whenthecracklengthincreasesabruptlybyAL increases:
UF'"(AL)2v(1-v).
(41)
ThecomparisonbetweenthepredictionbyusingEq.(41)andtheexperimentaldata,from
Ref.[57],is givenin Fig.3.Wecanseethat theoreticalcalculationsarein a goodagree-
mentwith experimentalevidence.The increasein UF(AL) mustbetakenintoaccount
whenpredictingthefunctionalityof actualstructuresmadeof materialswithmultifractal
microstructureandwhensimulatinggeodynamicphenomena.
5.3Fractalkineticsof ductileandbrittlefractureof solids
Fractureof solidsbelongto the classof processesin whichcomplexbehaviorat the
microscopiclevelisbehindthemacroscopiceffects.Behaviorof deformedsolidsubjected
tomechanicalactionin governedbytheformationandevolutionof dissipativestructures,
whichprovideoptimalconditionsfor dissipationof energyflowinginto thebodyfrom
theoutside.Manyfactorsplayrelevantrole:grain,boundaries,microcracks,particlesand
impurities,temperature,etc.,leadingto rich phenomenologyrangingfromcleavageto
ductilefracture.
Traditionally,theanalysisof processesthat controlfailureof solidsonmicrolevelhas
beenconfinedtotheconsiderationofmodelsthattakeintoaccountonlypairedinteratomic
bonds(seeRef.[10]).At thesametime,strongcorrelationoftherelativepositionofatoms
at distancesLo that significantlyexceedtheinteratomicdistancesrij,i.e.,thecorrela-
tionthat ensuresshearstabilityof solids,is characteristicof statesof condensedmatter.
Therefore,rheologicalbehavioris determinedby thedynamicsof collectiveexcitations
governedbyanexternalfactor[4],anda failureisacollective,essentiallynonequilibrium
FRACTAL SOLID MECHANICS
527
4.5
.
3
In UF
*
1.5
--
----
-- *
--
-----
o I.k-------
-10 -8
-6 -4 -2
O
InL
FIGURE 3.Dependence of surface fractal energy UF on the characteristic scale of fracture.Salid
lines are the results of calculation using Eq.(41) for m = 2 and v = 0.23 (curve 1) and v = 0.1
(curve 2).Points represent experimental data fram Ref.[57].
process,
whosekineticsisgovernedbyself-organizationofdissipativestructuresthaten-
sureoptimal(forspecified1pading)levelofdissipationof energyof theexternalaction[4].
Consistentallowancefor collectiveeffectsin fracturekinetics,includingeffectsat the
atomiclevel,is possiblewithintheframeworkof quantum-statisticalapproachwhichis
beingdeveloped[7],for synergeticsof deformedsolido
As aconsequenceof shearrigidityof condensedmedia,significantdifferenceappearsin
thecharacteristicrelaxationtimesfor theenergyTf,andimpulseTp«:Tf of atomsand
structuralelementsof thedeformedmedium[4,7].Therefore,in theinelasticdeformed
salid,mechanicalenergyaccumulatesinself-localized,highlynonequilibriumregionsthat
forroanopensubsystem(whichexchangesenergyandmatterwithregionsofthebodythat
arein aquasiequilibriumstate)in whichtheenergyof elasticdeformationsis dissipated.
It is preciselythiseffectthat givesrigeto thelocalizationof plasticstrainsandfracture
regionsin solids.It hasbeenshownelsewhere[31,50]that as a resultof the effectof
transversestrains,self-organizingdissipativestructuresduringirreversibledeformation
of themediawith shearrigidityhavescale-invariantmultifractalstructure.The fractal
dimensionof dissipativestructure,whichdeterminestherateof energydissipationupon
irreversibledeformationof a salid,correspondsto theeffectivevalueof thetransverse
straincoefficientlJeff,whichis thefunctionalof theprocessof irreversibledeformation.
Statisticalself-similarityof configurationsof cracksandthe multifractalnatureof the
fracturesurfacesof solids,whicharea sortof counterpartsto dissipativestructuresthat
528
A.S.BALANKINANDP.TAMAYO
aresuperposedon theoriginalstructureof thematerial,serveas graphicreflectionof
the scaleinvarianceof dissipativestructures[35].It has beenfoundthat well-defined
valuesof thefractaldimensionof crackprofilesandfracturesurfacearecharacteristicfor
differentsamplesofgivenmaterialthathasbeenputthroughanalogousthermomechanical
treatment[4,34].
In theprocessof irreversibledeformationof metals,theenergyaccumulatedin self-
localized,highlynonequilibriumregionsin theformof potentialenergyof elasticstrains
dissipatesthroughstressrelaxationduringplasticdeformation.It is consumedon the
formationof micro-,meso-,andmacrodefects,andconsequentlyis beingreleasedonthe
roughsurfaceof thecracksformed.If thecriticaldensityof thepotentialenergyof dilata-
tion (correspondingto thebifurcationpointof theoriginalstructureof thesolidbeing
deformed[49])Wev
=
u~/2Eor formchangingWed
=
Te/2G(whereE andG areYoung
modulusandshearmodulusTeanduearethecriticalshearstressandthecriticalstress
of microseparation),buildsup,self-organizationof thedissipativestructurethat ensures
subcriticalgrowthof a maincrackoccurs.Fractureduringsubcriticalgrowthof cracksis
related,in general,to thecooperationbetweentW()competingmechanismsgovernedby
differenttypesof dissipativestructuresthat areresponsiblefor crackformationthrough
microshearor microseparation[35,49].Everyeventof crackadvancementis associated
with theformationof critical nucleusby themechanismof microshear(if Wd~ Wed)
or microseparation(if Wv ~ Wev).The nucleiof cracksformedby microshearareasso-
ciatedwiththeattainment,in theslipplane,of thecriticaldensityof dislocations,and
thoseformedby microseparationareassociatedwith theattainmentof critical density
.
of disclinations in an element of the volume that has undergonethe extremeplastic
deformation.If failureis controlledbymicroshearassociatedwitha low-energy-intensity
pileupof dislocationsin theslipplane,localbrittleor quasibrittlefractureinitiatedby
translationalinstabilityoccurs.In thiscase,shearorcleavagefacesappearrevealedonthe
fracturesurface.But if microseparationassociatedwith high-energy-intensitypileupsof
disclinationsisthecontrollingmicromechanismof fracture,thenductilefractureinitiated
by rotationalinstabilityoccurs.A fractographicfeatureof local ductilefractureis the
presenceof raggedmicrorelief(undercyclicandstaticloadings)[49].
Fromtopologicalstandpoint,thedifferencebetweenductileandbrittlefractureisthat
in brittlefracturecrackfront,whosemetricdimensionlíesin therange
1
<
DF
~2,
(42)
is fractal(at stressesu ~Uca classicalGriffithcrackwitha smoothfront,whosemetric
dimensionequal to topological dimensionof the line dT
=
1 propagates),and in ductile
fracturethesurfaceof crackswhosedimensionis
2
<
DF
~3
(43)
isfractal.Thisdifferenceisattributabletothedifferenceinthekineticsofself-organization
of dissipativestructuresfromdefectsof differentmetricdimension.
To determinethepeculiaritiesof thekineticsof clusterizationof elementarydefects
responsibleforthetopologicaldifferencebetweenbrittle(quasibrittle)fractureandductile
FRACTAL SOLID MECHANICS
529
(quasiductile)fracture,let usexaminetheSmolukhovskiiequationwitha multiplicative
kernel<P(i,j)
rv
(i,j)w:
k k
d~k
=
L
<P(j,k - j)CjCk-j - Ck
L
<P(j,k)Cj,
j=l j
(44)
whereCj isconcentrationof clustersof j elementarydefects,andtheparameterwis the
fractiondimensionDf of thedissipativestructure:
w
=
D¡l
=
2+
()
2DF'
(45)
whereDF is metricdimensionof elementarydefects,and ()is the exponentof the anoma-
lousdiffusioncoefficientonthefractalD
rv
r-(J (seeRefs.[12-15]).
The metricdimensionof dislocationpileupsin theslipplane,asof elementarydefects
duringfailurethroughmicroshear,obviouslydoesnot exceedthetopologicaldimension
of theplane,i.e.,dHS 2.Therefore,in this casew 2:0.5,sinceit is alwaysin thecase
that ()
2:O(forpercolationclusters,DF =4/3andw=3/4>1/2).It isknown[15]that
in the casew
>
0.5 the asymptoticsolution of Eq.(44) describesthe growth of the only
"surviving"cluster,whichmaybeidentifiedwiththemaincrackin thecasein question.
Thedependenceof growthrateof themaincrackonits sizeR
dR
z
V
--rvR,
- dt
Dc
Z =1+(2w- I)Dc =1+(2+())dH- Dc,
(46)
hastheformof ParisequationV
rv
J{¡
rv
Rna
(seeRefs.[10,49]),wherea =0.5(2- DF) is
theexponentthatdefinestheasymptoticbehaviorofthestressintensityfactorJ{¡
rv
R-a
for crackswith fractalfront,andDe is thecorrelationdimensionof thecluster(main
crack).SettingZ
=na,wenave
[
Dc(2 +()) DF - 1
]
n
=
2 dH(2- dF) - 2- DF
.
(47)
It thusfollowsin thecaseDF =De <2,thatn >2(thisis in agreementwiththeresult
of numerousexperimentalstudies);here,if DF =dH=2[1- (1+())/(n- 2)],then,
n
=
2
[
1+ 1+
()
]
2
-
dH
'
m
=
n [1+2
~
dH]
,
(48)
wheremis theexponentin theWeibulldistribution(seeRef.[49]).
IngeneralcasewhereDF =De,thedimensionof thecrackfrontformedduringbrittle
andquasibrittlefractureis
DF
=
dH(n- 1).
2+()+(0.5n- l)dH
(49)
530
A.S.BALANKINANOP.TAMAYO
The metric dimensionof fracture surfaceis d~
=
2 in brittleandd~
=
DF
+1 > 2 in
quasibrittlefracture.
Thedimensionof defectsresponsiblefor rotationalinstability(disclinations,etc.)that
leadsto fracturevíathemicroseparationmechanismis dH>2.If thestrongercondition
dH > 2 +O,which ensuresthat (.¡)
<
0.5(i.e.,DF
>
2),is satisfiedhere,thenaccording
to Eq.(44),thereissimultaneousincreasein thesetof equalclusters(cracks)that form
themaincrackin accordancewith percolationmechanism[8,34].The dimensionof the
ductilefracturesurfaceis determinedby the fractionof energydissipatedas a result
of elasticstressrelaxationduringplasticdeformationandon theroughsurfaceof the
cracks(r¡).Ductile(quasiductile)fracturedueto rotationalinstabilityof thelatticeis
accompaniedbya cascadingof energytransferfromelasticdeformationsonlargerscales
Li+l to smallerscalesLi,clownto themicroscaleLo,wheretheresidualenergyof elastic
deformationsgoesto theformationof newdiscontinuitysurface(theprocessis similarto
cascadebreakupof vorticesduringtheturbulenceof fluidflows).If thefractionof energy
of elasticdeformationsconsumedin dissipativeprocessesuponthetransitionfromone
structuralleveltoanotheris independentof Li (i.e.,if r¡
=const),thenthelawofenergy
conservation
(1- r¡)NiWcvt:J.Ri
=
Ni+l Wcvt:J.Ri+l,
whereNi is thenumberof fragmentsof thecrackof thei-th scaleLi andt:J.Riis the
increasein thelineardimensionalityof a crack,yieldstherelation
dS
- 2 In(1- r¡)
F - -
1 ~2,
nI\;
1\;
=
Li+1/Li.
(50)
Here 1\;
=
constis theself-similarityparameter,whichdefinesthehierarchyof spatial
scalesof thestructurallevelsof failure.Sinced~~3,wehaver¡ ~1- 1\;-1.At larger¡,
fractureis impossible.
.
In quasielasticfracture,theprimarymechanismofdissipationistherelaxationofelastic
stressesontheroughfractalcrackCace.Herether¡-thpartofenergyofelasticdeformations
isdissipated(theremaindergoestotheformationofnewsurfacesduringcrackformation):

=
1- 1\;-/3
,
j3
=d~- (d- 1)=dF- 1>O.
During propagationof classical Griffith's crack,we havej3
=
Oand r¡
=
Oin agreement
withtheclassicalresulto
Mechanismsof fractureof solidsimitatedin thestochasticmodel(26),(27),of fracture
of elasticlatticeforvariousvaluesof m-parameterandtopologicallyequivalentclassesof
thekineticandpercolationmodelsof fractureof solidsarelistedin Table
IV.
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