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10
T.Derda
[1-3].TheFBMisastatisticalapproachformodellingandanalysisofstochastic
fracture-failureprocessesindisorderedmaterialssubjectedtoexternalload[4-6].
Therestofthechapterisorganisedasfollows.Section2presentstheapplied
model.Section3discussestheresultsobtainedforquasi-staticandsuddenloading
procedures.Weanalysecriticalloadsandtheprobabilitiesofsystemsurvival.
ThechapterendswithashortConclusionsection.
2.Descriptionofthemodel
ConsideracollectionofN
±X
LL
verticalpillarslocatedatthenodesofthe
squarelattice.Lisalinearsystemsize.Thepillarsarefixedtoaflatsubstrate.
Therandomnessofthesystemisreflectedinpillar-strength-thresholds,
σ
i
th
,
i
±
1,2,..,
N
.
Thestrength-thresholdsarequenchedrandomvariablesdrawnfrom
agivenprobabilitydistribution.Usually,theuniformdistribution,withdensity
pσ
(
th
)
±
1
andcumulativedistributionfunction
Pσ
(
th
)
±σ
th
,
servesasastarting
pointfortheanalysis.Therangeofthefunctionisbetween0and1.Thedistribution
withamuchbetterphysicalfoundationistheWeibulldistribution[7,8]withdensity
p
pX
,
(
σ
th
)
±
p(
X
|
k
σ
X
th
N
|
)
p-
1
exp
(
|-
|
k
(
|
k
σ
X
th
N
|
)
p
N
|
|
)
andcumulativedistributionfunction
P
pX
,
(
σ
th
)
±-
1exp
(
|-
|
k
(
|
k
σ
X
th
N
|
)
p
N
|
|
)
(1)
(2)
wherepandXaretheWeibullindex(shapeparameter)andscaleparameter,
respectively.Parameterpcontrolstheamountofdisorderinthesystem.Weibull
distributionisatypeIIIextremevaluedistributionandrelatestominima.Two
otherextremevaluedistributionsareGumbelandFréchetdistributions[9].Inthe
caseofGumbeldistributionfar-outtailsreachintophysicallymeaninglessnegative
strengths.TheFréchetdistributionisboundedfrombelowandhasaheavyupper
tail(slowlyconvergingto1)–thisisincontrasttotheWeibullcasewhichhas
notail.ComparisonbetweentheFréchetandWeibulldistributionsisgraphically
reportedinFigure1.Inthiswork,weassumethatpillar-strength-thresholdsare
Fréchetdistributedrandomvariables[10].Theprobabilitydensityfunctionofthe
Fréchetdistributionisgivenby
p
m
,
Y
(
σ
th
)
±
m
Y
(
|
k
σ
Y
th
N
|
)
--
m
1
exp
(
|
|
k
-
(
|
k
σ
Y
th
N
|
)
-
m
N
|
|
)
(3)
