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RESEARCHOpenAccessEffectoftheloadingdurationonthelinearviscoelasticparametersoftropicalwood:caseofTectonagrandisL.
f(Teak)andDiospyrosmespiliformis(Ebony)ofBeninRepublicAgapiKocouviHouanou1*,AdolpheDèfodjiTchéhouali1andAmosErickFoudjet2AbstractJudiciousandregulateduseofwoodasabuildingmaterialisbetterthanthatofmanyotherconventionalmaterialsintermsofenvironmentalissuesofthiscentury.
Thestudyofthebehaviorofwoodrequiresabetterunderstandingofthecharacteristicsindifferentpossiblecasesofloadingincludingloadsappliedinstantly,loadsappliedforashorttimeandloadsappliedforalongtime.
Thepurposeofthisstudyistoevaluatetheinfluenceoftheloadingdurationonthelinearviscoelasticparametersoftropicalwoodincreeptest.
Creeptestsconductedontwospeciesoftropicalwood,TectonagrandisL.
fandDiospyrosmespiliformis,werecarriedoutforatotalloadingdurationof15hoursbysubjectingsamplestobendingtestthroughwithequalstraininallsections.
Aftermeasuringtheinstantaneousdeflection,theothermeasurementswerecarriedoutatregulartimeeach30minutes.
Eachrecordeddeflectionwasconvertedintolongitudinaldeformationandthedatawereanalyzedbyconsideringfourteenloadingdurations.
Usingtheleastsquaresmethod,thedynamicmodulusofelasticityandthemodulusofdynamicviscosityweredeterminedforeachloadingtime.
Theresultsshowedthattheloadingtimehasnoinfluenceonthemodulusofdynamicviscosity.
Ontheotherhand,thedynamicmodulusofelasticitydecreasesandtendstowardszero.
Goodagreementbetweencreeptestdataanddynamicmodulusofelasticitywasfoundusingmathematicalfunctioninpower.
Suitably,the"power"functionestablishedbetweentheelasticdynamicmodulusandtheloadingdurationcanbeusedtoextrapolatedeformationsvalues.
Keywords:Tropicalwood;Bendingcreep;Loadingduration;Dynamicmodulusofelasticity;DynamicmodulusofviscosityIntroductionWoodusedasabuildingmaterialisanaturalresourcewithmultiplebenefits,includingquotingforexample,thesignificantreductionofthenegativeenvironmentalimpactsgenerallyregisteredwhenusingothercurrentbuildingmaterials.
Itsuseinthefieldofcivilengineeringdatesforaverylongtimeandisencounteredintheconstructionoflargestructuressuchashouses,bridgesetc.
Deconstructionwastegeneratedbythedemolitionofthewoodenstruc-tures,attheendoftheirlifearerareeasymanageablewaste.
Dependingonthedestination,thewoodenstruc-turesmaybesolicitedbyshortloadingduration,averageloadingdurationorlongloadingduration.
Deformationsunderthesechargesconsistofinstantan-eousdeformationsanddeferreddeformations.
Thoseundershortloadingdurationarecontrollableandtakenintoaccountwhendesigningclassicworksbutthecontrolofthefailuremechanismresultingofdeferreddeforma-tionsrequiresknowledgeoftheviscoelasticparameters.
*Correspondence:agapikh13@yahoo.
fr1LaboratoryofEnergyandAppliedMechanics-PolytechnicSchoolofAbomey-Calavi,UniversityofAbomey-Calavi,Abomey-Calavi,BeninRepublicFulllistofauthorinformationisavailableattheendofthearticleaSpringerOpenJournal2014Houanouetal.
;licenseeSpringer.
ThisisanopenaccessarticledistributedunderthetermsoftheCreativeCommonsAttributionLicense(http://creativecommons.
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0),whichpermitsunrestricteduse,distribution,andreproductioninanymedium,providedtheoriginalworkisproperlycited.
Houanouetal.
SpringerPlus2014,3:74http://www.
springerplus.
com/content/3/1/74Severalstudieshavebeenconductedtodemonstratetheviscoelasticbehaviorofwoodinordertoestablishitsmainproperties.
Thus,tocharacterizethedeferredbehav-iorofwood,mostofauthorsusecreeptestsattheexpenseofrelaxationtests,becauseoftheconsistentloadingmodewiththesolicitationconditionsencounteredincurrentus-agesofwoodandeasyimplementationofexperimentaldevicesfordeflectionmeasurementswithaviewtocalcu-latingthecorrespondingdeformations.
Theexplanationofthelinearrangeofwoodviscoelasticbehaviorwasalsosubjecttoseveralinvestigations.
Asre-portedbyMontero(2010),studiesconductedbyKingstonandClarke(1961),NakaiandGrossman(1983)andMukudai(1983)showedthatthewoodviscoelasticityislinearforlowerloadingsat40%oftheultimatetensilestrengthandtheBoltzmannsuperpositionprincipleisvalidaccordingtoNakaiandGrossman(1983).
Whilestudyingthetimberrheology,Foudjet(1986)alsostudiedthelinearityofitsviscoelasticbehaviorfromcreeptestsonsometropicalspecies(Azobé,Tali,SapelliandMovingui).
Theobtainedresultsshowthatthede-ferredbehavioroftimberisviscoelasticlinearforstresseslessthanorequalto35%oftheultimatetensilestrength.
Thesemeasurementswereobtainedoniso-stresssamples(cantilever)stabilizedat12%ofmoistureandsubmittedlongitudinallyatdifferentlevelsofconfin-ingstress(respectively25%,30%,35%and42%oftheul-timatetensilestrength).
AsforRandriambololona(2003),hereportedinhispaperthatthelinearitylimitofviscoelasticbehaviorofwooddependsonthetypeofstressandissituatedatastressbetween10%and20%oftheultimatetensilestrengthwhenthetestiscarriedoutincompressionandbetween20%and30%oftheultimatetensilestrengthwhenitisthebendingortensilecreeptest.
Placet(2006)reportedinhispaperthattheviscoelasticbehaviorofwoodisstronglyinfluencedbytemperatureandmoistureinadditiontotheloadingdurationinre-spectofthepolymericnatureoftheseconstituents.
In-deed,hefoundoutthatthebehaviorofaviscoelasticmaterialtohighertemperatureforshortloadingtimesisequivalenttothatofthesamematerialatlowertemperature,butforlongertimes.
Thisistheprincipleoftime-temperatureequivalenceortheprincipleoftime-temperaturesuperposition(Dlouhà2009;Placet2006).
InthepaperofMontero(2010),itissaidthatwhenthewatercontentisbelowthefibersaturationpoint,itinfluencestheviscoelasticbehaviorattwolevels:thekinetics'evolutionandmecanosorptiveeffectduetotheviscosityofwoodwhichdependsonitswatercontent,butisalsoverysensitivetothevariationofthewatercontent.
Thus,theviscoelasticcomplianceisseventimeshigherforwetcreep(moisture22%),thesamewithadrycreep(moisture0,5%).
Asformecanosorptivefact,thefirstworkswerepub-lishedin1960byArmstrongandKingstonhighlightingtheinfluenceofthevariationofmoistureinwoodinitsdeferredbehavior.
Inbendingtests,theycomparethecreepofwoodsampleskeptatconstantmoisturewiththeonethatcandryduringthetest.
Fromtheseresults,itap-pearsthatthecreeponthesesamplesisatleasttwotimesgreaterthantheonesmaintainedatconstantmoisture.
Ayearlater,ArmstrongandChristensen(1961)bydetailingpreviousstudieshaveindicatedthatthisincreasedependsontherateofsorptionandnotthemoistureoftheloadedsample(Montero2010).
Theseresultsonmeca-nosorptionpavedthewayforotherstudies,includingthoseofRandriambololona(2003)devotedtomodelingthedeferredbehaviorofwoodinavariableenvironment.
Theviscoelasticmodelisbyfarthemostwidelyusedformodelingthemechanicalbehaviorofwood.
Infact,thelinearviscoelasticbehaviorisrepresentedgenerallybyconstructingamodelconsistingofanassemblyofspringsanddashpots.
Itisthereforeananalogandsym-bolicmodelrepresentedbyacombinationofspringsanddashpotsinseriesandinparallelmoreorlesscom-plex(Foudjet1986;Placet2006;Dlouhà2009).
TheworksofHaqueetal.
(2000),devotedtothecomparisonoftherelevanceofthesedifferentmodelsplusanempir-icalmodelbasedontheequationofBailer-Nortonem-piricalmodel,itwasfoundthattheKelvinmodelseemstobethebestsuitedtointerpolatetheexperimentalcurves(Moutee2006;Husson2009).
Thus,severalau-thorsadopttheKelvin'smodel,ormorepreciselytheseriesconnectionofnKelvinelementstorestoretheviscoelasticbehavioroftimberinacreeptest.
However,identifyingproblemsquicklybecomeinsolublebecauseoneneedstodetermineatleastasmanycoefficientsaselementsintroduced,whichmaybeunworkableespe-ciallyinpracticejustforthefactthattheseparametershighlydependonthemoistureandtemperature.
Inthiscontext,Foudjet(1986)showedarheologicalmodelwithmaximumtwo(2)Kelvin-Voigtelementsconnectedinserieswhichwaswidelyenoughtorepresenttheownlinearviscoelasticbehaviorofwood.
Otherstudiescarriedoutonpolymerslinearviscoelas-ticbehaviorhaveshowedthatcreepcomplianceJ(t),isonlyafunctionoftimeandnotafunctionofthemagni-tudesofstressandstrain(Bower2002;BrinsonandBrinson2008;ChandaandRoy2009)andthedeform-ation(strain)dependontheappliedstress(Barnersetal.
1993;ChandaandRoy2009).
Consideringtheresultsobtainedbythesepreviousworks,thestudiespublishedbyHouanouetal.
(2012)weredevotedtotheidentificationoflinearviscoelasticparemetersoftwotropicalwoodsatagivenmoistureandconstantlyheld,underasteadyappliedloadduringtheentiretestperiod.
Houanouetal.
SpringerPlus2014,3:74Page2of12http://www.
springerplus.
com/content/3/1/74AsforEurocode5(1995),woodisclassifiedaccordingtothemechanicalstrengthcriteriadefinedbytheruleswhichensurethereliabilitywithregardtotheuseforwhichitisintended.
Indeed,thisstandarddefinesacoeffi-cient(kmod),amendingthestrength,takingintoaccounttheclassofserviceofloadingdurationandmoistureofthewood.
Thiscoefficient(kmod)isusedtodeterminethedesignvalueXdofapropertyofthematerial.
XdkmodXkγm1Where,Xk:characteristicvalueofmaterial'spropertyγm:partialcoefficientapplyingtotheproperty.
Theaimofthisstudyistodeterminethemechanismtobetakenintoaccountoftheeffectoftheloadingdur-ationinthedesignofawoodenstructureinthelinearviscoelasticareaandnotsimplyintheelasticareaasthingsweredoneuntilnow.
Moreprecisely,theaimofthisworkistostudythein-fluenceofloadingdurationonthelinearviscoelasticpa-rameters(dynamicmodulusofelasticityanddynamicmodulusofviscosity),woodmoistureandappliedloadbeingmaintainedconstantthroughoutthecreeptest.
Otherwise,itwillbetomodelthebehaviorofeachpar-ameterasafunctionofloadingdurationtoextrapolatethevaluesofcreepordeformation.
Finally,itwillhelptodeducethelengthofabendingcreeptesttwo(2)pointsafterwhichaKelvin-Voigtmodelreflectsoptimallythelinearviscoelasticbehaviorofwood.
Thestudywilliden-tifyamodelforpredictingthemechanicallapseofanelementofstructuresubjectedtosuchstressesandcon-sequentlywilldevelopasuitablemethodtoextrapolatelongitudinalstrainvalues.
Also,thisworkswillpermittodeterminethespecieswhichhasthebestlinearvisco-elasticparameters.
Toachievetheseobjectives,weusedtheparameteridentificationapproachdescribedinHouanouetal.
(2012).
Thisapproachisappliedbyfollowingtheobservationwindowscreepcarefullychosenandappropri-atemathematicalfunctionstobuildtheextrapolationmodels.
MaterialandmethodsThesamplesweretakenfromthesameboardoftheheart-woodfollowingthelongitudinaldirection(Houanouetal.
2012).
Foreachspecies,twelve(12)experimentalspeci-mensweremadeupasshowninFigure1.
Thecutupsamplesprovidetheshapeofanequallysolicitedbeaminallitssections.
Theexperimentalspec-imensweredriedto12%moisturecontentinamoderndryer,inaccordancewiththenormalconditionsoftemperature,pressureandspeeddrying.
Theywerecare-fullysurroundedbyaluminumfoilinordertomaintaintheirwatercontentundercontrol.
Thebendingcreeptestconsistsonsubjectingthesampletotwopointsbendingtest.
Thesamplesareem-beddedatoneendand20%ofthebendingfailureload(let19.
4MPaforTeakand25.
2MPaforEbony)wasappliedattheotherend.
Thisloadisappliedto300mmfromtheotherendofthebeam(Figure2)(Houanouetal.
2012).
Creeptestwerecarriedoutwithinatotaldurationof15hours.
Thedeflectionsweremeasuredbymeansofacomparatorwithanaccuracyof1/100mmevery30minutesatmid-spanofthebeamaftermeasuringtheinstantaneousdeflection.
Datahavebeentreatedbyconsidering14periods:(0-2h),(0-3h),(0-4h),(0-5h),(0-6h),(0-7h),(0-8h)(0-9h),(0-10h),(0-11h),(0-12h),(0-13h),(0-14h)and(0-15h).
Eachperiodrepresentsanobservationwindow.
Thespecimenswereweighedatthebeginningandattheendofthecreeptest.
Thetemperatureismaintainedataconstantvalueduringmanipulation.
Thedeflectionsrecordedhavebeenconvertedtolongitudinaldeform-ation(Houanouetal.
2012)usingthefollowingformula:ε4σucσutσucσut2ff2L2h2Where,Figure1Configurationofspecimens.
Houanouetal.
SpringerPlus2014,3:74Page3of12http://www.
springerplus.
com/content/3/1/74σuc:ultimecompressivestress(MPa)σut:ultimetensilestress(MPa)f:beamdeflection(mm)L:beamspan(mm)h:heightofthebeam(mm)Thesearrowswereusedtocalculatetheviscoelasticcreepcompliance,J(t),usingthefollowingformuladerivedfromFoudjet(1986):Jtεσ03Where:εisthedeformationcalculatedusing(2),σ0meanstestloading.
CreepcomplianceJ(t)isthesumoftheinstantaneouscreepcomplianceJ(τ)andthelinearviscoelasticcreepcomplianceJ(t>τ)withτ,thetimeatwhichtheinstant-aneousdeformationisread(τ=15seconds).
Themathematicalexpressionofcreepcomplianceisoftheform(Foudjet1986;Guitard1987;Houanouetal.
2012):Jtεtσ01E01E1expEηtwitht>04Equation4isderivedfromtherheologicalmodelofZener.
ThismodelistheseriescombinationofaspringcharacterizingtheinstantaneousdeformationandKelvin-Voigtmodelwhichrepresentstheowncreepofwoodinthelinearviscoelasticdomain(Foudjet1986;Guitard1987;Houanouetal.
2012).
Inthisexpression,"E0"isHookeelasticitymodulus;"E"meansthedynamicelasticitymodulusofthespringand"η"thedynamicviscositymodulusofthedamper.
Foridentification,wehave:Jτ1E05andJt1E1expEηtwitht>τ6Foreachobservationwindow,theoptimumvaluesofthe"E"and"η"oftheexpressionofcreepcomplianceinthelinearviscoelasticfield(6)aredeterminedbyadjust-ingtheowncreepcomplianceofusingthemethodofleastsquaresnonlinearasdescribedinHouanouetal.
(2012).
Theowncreepcomplianceiscalculatedwith(3)usingthedelayeddeformation.
Forabetterleadingofdataanalysisinpurelylinearviscoelasticfield,abasechangeismadewheretheoriginisat(0,0)andthestartingpointoftheexperimentisnowat(τ;-j(τ))inthenewcoordinatesystem.
Thischangeallowsustoaccountfortheperiodofreadingtheinstantaneousdeformationcharacterizingthepurelyelasticrange.
Themathematicalmodeltopredicttheevolutionofthedynamicmodulusofelasticitywiththeloadingdurationwasestablishedbymeansoftheleastsquaresmethodandthefittingequationdeductedfromtheexperimen-talcurvescanbeexpressedasfollow(PolyaninandManzhirov2007):Etatbwitha>0;b07Etaebtwitha>0;b0;b0theexponentialfunctionE(t)=aebtwitha>0;b<0andt≥0.
Eachoftheselawscreatesapackagethattendstozerowhentheloadingtimeisrela-tivelylong.
However,thepowerlawismoreappropriate.
Further,theresultsshowedthatonecancharacterizethefreecreepofwoodtroughtheassessmentofthelinearviscoelasticbehavioradoptingtherheologicalmodelofKelvin-Voigtinbendingcreeptestwhenitiscarriedoutwithinamaximumloadingdurationof9hours.
Finally,studieshaveshownthatforsomeshortloadingtimesinferiorto2h42mn,thelinearviscoelasticpa-rametersofEbonyareslightlybetterthanthoseofTeak.
Butbeyondthat,thelinearviscoelasticparametersofTeakarefranklybetterthanthoseofEbony.
CompetinginterestsTheauthorsdeclarethattheyhavenocompetinginterests.
Authors'contributionsHKA,TDAandFAEconductedastudyontheinfluenceoftheloadingdurationonlinearviscoelasticparametersoftwotropicalspecies(teakandebony)ofBeninRepublic,participatedintheanalysisandexplanationofthedatacollectedanddraftedthemanuscript.
Authordetails1LaboratoryofEnergyandAppliedMechanics-PolytechnicSchoolofAbomey-Calavi,UniversityofAbomey-Calavi,Abomey-Calavi,BeninRepublic.
2CRESAForest–Wood,UniversityofDschang,Dschang,Cameroon.
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:Effectoftheloadingdurationonthelinearviscoelasticparametersoftropicalwood:caseofTectonagrandisL.
f(Teak)andDiospyrosmespiliformis(Ebony)ofBeninRepublic.
SpringerPlus20143:74.
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