Avoiding waviness of relaxation spectra.pdf

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1、RheolActa(2009)48:709713DOI10.1007/s00397-009-0370-zRAPIDCOMMUNICATIONAvoidingwavinessofrelaxationspectraJoachimKaschta·FlorianJ.StadlerReceived:22August2007/Accepted:18May2009/Publishedonline:2June2009©Springer-Verlag2009AbstractSpectraarematerialfunction

2、s,whosedif-Determinationofrelaxationspectrafromoscillatoryferentintegraltransformsyieldthemeasurablerhe-shear—stateoftheartologicalquantitiesinthelinearviscoelasticregime.Therefore,theirknowledgeisofhighfundamentalin-Asthedynamicmechanicalexperimentisthemo

3、stterest.However,thecalculationofspectrafromex-widelyusedrheologicalexperiment,werestrictour-perimentaldataisanill-posedproblem.Thus,itisselvestothedeterminationofrelaxationspectrafromhamperedbytheproblemthatatoolowdensityofoscillatoryshearwithoutlossofgen

4、erality.Therealandrelaxationmodesdoesnotleadtoagooddescriptionimaginarypartsofthecomplexmodulusdeterminedinoftheinputdata,whileahigheroneusuallycausesaafrequencysweeparelinkedtothediscreterelaxationwavyspectrumwhichcannotbeinterpreted.Toover-spectrumofnpai

5、rsofa(positive)relaxationtimeτiandcomethisproblem,anadditionalcriterionassuminga(positive)relaxationstrengthgiby:onlygradualchangesinthespectrumisintroducednω2×τ2allowingforanincreaseinmodedensitywithoutaniG(ω)=gi(τi)×22(1)enhancedwavinessofthespectrum

6、.Thisisnovelin1+ω×τi=1icomparisontopreviouslypublishedspectraalgorithmsandcommercialsoftwarepackages.nω×τiKeywordsRelaxationspectrum·Errorminimization·G(ω)=gi(τi)×22(2)1+ω×τi=1iSpectrumshapewithωbeingtheangularfrequencyoftheexperiment.Allalgorithmsfor

7、thesimultaneousinversionofEqs.1and2published(BaumgärtelandWinter1989;TschoeglandEmri1992,1993;EmriandTschoegl1993;KaschtaandSchwarzl1994a,b)usetheleastrelativemeansquareerrorDasacriteriontojudgethequal-J.Kaschta·F.J.Stadlerityofthedescriptionoftheinputdata

8、bythespec-InstituteofPolymerMaterials,DepartmentofMaterialstrumdetermined.InthecaseofadynamicmechanicalScience,Friedrich-Alexander-UniversityexperimentconsistingofNmeasureddatapointsofErlangen-Nürnberg,Marten