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Chapter1:AxiallyloadedarraysofpillarswithFréchet-distributedstrengths
andcumulativedistributionfunction
P
m
,
Y
(
σ
th
)
±
exp
(
|
|
k
-|
(
k
σ
Y
th
N
|
)
-
m
N
|
|
)
wheremandYareshapeandscaleparameters,respectively.
11
(4)
Fig.1.ProbabilitydensityfunctionsfortheFréchet(leftpanel,γ=1)
andWeibull(rightpanel,λ=1)distributions
Duringtheloadingprocess,eachcomponentischaracterisedbyoneoftwo
states:working(whenthelocalloadonthepillardoesnotexceeditsstrength-
-threshold)orfailed(otherwise).Wealsoassumethatthepillarsarenotrepairable.
Thekeyaspectofthemodelisamechanismofloadtransferfromthebroken
pillarstotheintactones.Wecandistinguishtwoextremalrangesofloadtransfer:
globalloadsharing(GLS)andlocalloadsharing(LLS).IntheGLSscenario
theloadcomingfromdestroyedcomponentisequallyredistributedtoallreaming
intactelementsinthesystem.Hence,thisrulerepresentslong-rangeinteractions
andaperfectlystiffsubstrate.ThisisincontrasttotheLLSrule–inthiscasepil-
larsarelocatedonanonrigidsurfacethathasanonvanishingcompliance.TheLLS
rulecorrespondstoshort-rangeinteractions–theloadfromthedestroyedcompo-
nentistransferredonlytoitsnearestintactneighbours.InthecaseoftheLLS
model,theanalyticalsolutionsaredifficulttoachieveduetonon-trivial,spatial
correlations[5].TheGLSmodelsrepresentmean-fieldapproximationsandcanbe
solvedanalytically.
Thepillararrayissubjectedtoaxialexternalloading.Inoursimulationswe
considertwodifferentloadingprocedures:quasi-static(gradual)loadingandsudden
loading[11].Regardlessofthetypeofloading,itisnotedthattheexternalloading
isalwaysuniform,however,theinternalloadtransferscanleadtoinhomogeneities
