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1、MECHANICSOFFLEXIBLEFIBERASSEMBLEISEditedbyJOHNW.S.HEARLE,JOHNJ.THWAITESANDJAFARGHOLIAMIRBAYATThebendingofyarnsandplainwovenfabricsP.GrosbergDepartmentoftextileindustries,theUniversityofLeeds,EnglandAbstractThebendingofyarnsisanalyzedintermsoftheindependentbendingofasetofhelices.R
2、easonsfordeviationfromthismodelareconsideredandtheeffectoffrictiononbendingbehaviorofyarnsisconsidered.Itisshownthatforbendingoffabricstheeffectofsterichindrancetothebendingoftheyarnmustalsobeconsidered1.IntroductionThebendingoffiberstructuresdiffersmarkedlyfromthebehaviorofsolid
3、structuresinthatbyandlargethefibersinthestructurebendindependentlyofeachothersothattheresistancetobendingofthestructurebecomesthesumofthebendingresistancesofthefibersandisnotproportionaltothemomentofinertiaofthecrosssectionofthestructure.Thustoafirstorderofaccumulationthebendingr
4、esistanceofayarnisproportionaltothenumberoffibersinitscrosssectionwhichisproportionaltothesquareoftheyarnradius,R2,whilethemomentofinertiaoftheyarncrosssectionisproportionaltoR4.Consequentlyitispossibletouserelativelythickyarnswithoutproducingexceedinglyrigidyarns.Thebendingofaya
5、rnorfabricisusuallydescribedintermsofitsflexuralrigidity,B,definedbytheproductofthebendingcouple,M,andtheresultingradiusofcurvatureρ.IfthestructurebendingbehaviorislinearMisdirectlyproportionaltothecurvature,i.e.1/ρ,andBisaconstant.Theanalysisofthebendingofyarnshasthereforeusuall
6、yprocessedbythedeterminationofthebendingresistanceofthefibersintheyarn,thesefibersbeingassumedtolieinsimplehelices.2.THENBENDIGNOFAYARNThesimplestanalysisofthebendingofafibershelixisthatduetoliveseyandowen,1964.Theyusedwhatisinessenceacostiliagnotreatmentinwhichitisassumedthatthe
7、deformedgeometryispracticallyidenticalwiththeinitialgeometry.Ifthereforeahelixofradius,r,andhelixangle,θ,isdeformedbyacouple,M,it1isrelativelyeasytoshowthatifΦistheanglemadebythehelixradiustoapointonthehelixandtheplaneofbending,thenbendingandtensionalcouplesareapplied122atthispoi
8、ntequaltoM(1−sin∅sinθ)2andMsin∅sinθrespe
