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SelectedTopicsinContemporaryMathematicalModeling2021
CHAPTER1:AXIALLYLOADEDARRAYSOFPILLARS
WITHFRÉCHET-DISTRIBUTEDSTRENGTHS
TomaszDerda
DepartmentofMathematics,CzestochowaUniversityofTechnology
Czestochowa,Poland
tomasz.derda@pcz.pl
Abstract.Multicomponentsystemsarecommonlyencounteredinnano-scaletechnology.
Specifically,nanopillararraysbelongtoabroadclassofmulticomponentnanodevices,
whicharecomposedofalargenumberofalmostidenticalpartsthatworkasaunit.Within
aFibreBundleModelframework,wesimulatedtheloadingprocessesintwo-dimensional
arraysofverticalpillars.Thepillarsarefixedtoaflatsubstrate.Thestiffnessofthesub-
strateisresponsiblefortheeffectiverangeofinteractionbetweenpillars.Weexplorelocal
andgloballoadsharingrulesthatcorrespondtoshortandlongrangeinteractions.Theload-
ingprocessisdoneintwoways,i.e.quasi-staticallyandbyapplicationofsuddenloading
procedure.Theimportantelementofthemodelisdistributionofpillar-strength-thresholds–
inthisworkweassumethatstrength-thresholdsaredistributedaccordingtoFréchetdistri-
bution.Basedonintensivecomputersimulations,weanalysesuchcharacteristicslikecriti-
calloads,includingtheirdistributions,andsurvivingprobabilities.
Keywords:fibrebundlemodel,multicomponentsystems,extremevaluedistribution,surviv-
ingprobability
1.Introduction
Reliabilityofmulticomponentsystemsisacommonconcernofmanufacturers
andcustomers.Inthecaseofamulticomponentsystem,acrucialissueishow
performancesofindividualcomponentscombineintogeneralperformanceofthe
systemconsistedofthesecomponents.Anexampleofmulticomponentsystemsare
arraysofnanopillars,whichplayanimportantroleinnanotechnology.Forexam-
ple,nanopillararraysareemployedinphotovoltaicdevicesorgridcellsusedin
experimentalbiomedicine.
Weassumethatthepillarsinthearrayaregeometricallyalmostidentical,but
differintheirinternalstructurethatleadstodifferentiationofpillarstrengths.Hence,
oncethesystemissubjectedtolowormoderateexternalloadsomecomponents
mayfail.Initiatedbythesefailures,thedestructionprocessmaybeself-sustained
inaformofavalanchesofsimultaneouslydamagedcomponents.Toeffectively
studyfailuresinmulticomponentsystems,weemploytheFibreBundleModel(FBM)
