Uncertainty on Fatigue Damage Accumulation for Composite Materials Toft, Henrik Stensgaard; Sørensen, John Dalsgaard

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1 Aalborg Unverstet Uncertanty on Fatgue Damage Accumulaton for Composte Materals Toft, Henrk Stensgaard; Sørensen, John Dalsgaard Publshed n: Proceedngs of the Twenty Second Nordc Semnar on Computatonal Mechancs Publcaton date: 009 Document Verson Publsher's PDF, also known as Verson of record Lnk to publcaton from Aalborg Unversty Ctaton for publshed verson (APA): Toft, H. S., & Sørensen, J. D. (009). Uncertanty on Fatgue Damage Accumulaton for Composte Materals. In L. Damklde, L. Andersen, A. S. Krstensen, & E. Lund (Eds.), Proceedngs of the Twenty Second Nordc Semnar on Computatonal Mechancs (pp ). Department of Cvl Engneerng, Aalborg Unversty. DCE Techncal Memorandum, No. General rghts Copyrght and moral rghts for the publcatons made accessble n the publc portal are retaned by the authors and/or other copyrght owners and t s a condton of accessng publcatons that users recognse and abde by the legal requrements assocated wth these rghts.? Users may download and prnt one copy of any publcaton from the publc portal for the purpose of prvate study or research.? You may not further dstrbute the materal or use t for any proft-makng actvty or commercal gan? You may freely dstrbute the URL dentfyng the publcaton n the publc portal? Take down polcy If you beleve that ths document breaches copyrght please contact us at vbn@aub.aau.dk provdng detals, and we wll remove access to the work mmedately and nvestgate your clam. Downloaded from vbn.aau.dk on: jun 4, 08

2 Proceedngs of the Twenty Second Nordc Semnar on Computatonal Mechancs Aalborg Unversty 009 ISSN DCE Techncal Memorandum No. Uncertanty on Fatgue Damage Accumulaton for Composte Materals Henrk Stensgaard Toft* and John Dalsgaard Sørensen Department of Cvl Engneerng Aalborg Unversty, Aalborg, Denmark emal: Summary In the present paper stochastc models for fatgue damage accumulaton for composte materals are presented based on publc avalable constant and varable ampltude fatgue tests. The methods used for estmatng the SN-curve and accumulated fatgue damage are presented. Introducton Damage accumulaton models for composte materals exposed to fatgue loadng have been wdely consdered n the lterature, see e.g. [] for a revew. However, even though new emprcal and physcal models for the accumulaton of damage are proposed, these models do not seem to perform much better than the lnear damage accumulaton proposed by Mner []. For ths reason the accumulated damage s normally determned by Mners rule as recommended n [3] for wnd turbne blades. The uncertantes n damage accumulaton based on Mners rule can n general be dvded nto three parts: Physcal uncertanty on the SN-curves Statstcal uncertanty on the SN-curves due to a lmted number of tests Model uncertanty on Mners rule The physcal uncertanty on the SN-curves s due to the natural nherent uncertanty n the materal and can not be reduced. The statstcal uncertanty can be reduced by performng addtonal fatgue tests and the model uncertanty can n prncple be reduced by adoptng a better model. In the present paper s the physcal and statstcal uncertanty on SN-curves for composte materal determned based on a number of constant ampltude fatgue tests performed wth dfferent mean stresses. Based on varable ampltude fatgue tests for the same materal usng a standard load spectrum and the estmated SN-curves based on constant ampltude fatgue tests s the model uncertanty on Mners rule determned. In most standards and regulatons ncludng [3] are fatgue desgn performed by usng a determnstc desgn approach. However, calculaton of the accumulated fatgue damage ncludes sgnfcant uncertantes for whch reason a probablstc desgn approach should be adopted n order to take the ndvdual uncertantes nto account n a ratonal manner. The stochastc models determned n the present paper forms the bass for a probablstc modelng of the consdered class of composte materals n fatgue loadng. Constant ampltude fatgue tests The constant ampltude and varable ampltude fatgue tests used n the present paper are gven n the OptDAT database [4] for geometry R04 MD (MultDrectonal lamnate). Ths geometry has been selected due to the many fatgue tests performed wth ths geometry. For composte materals the mean stress can have a sgnfcant nfluence on the fatgue propertes whch can be taken nto 53

3 account by calculaton of an SN-curve for dfferent R-ratos and arrangng these n a constant lfe dagram. The R-rato s defned by: mn R () max where mn and max are the mnmum and maxmum stress n a stress cycle respectvely. Dfferent types of SN-curves have been used for composte materal, but no specfc SN-curve have been recommended n [3]. In the present study s a log-log SN-curve used: log N log K mlog () where N s the number of cycles to falure, s the stress range and s parameter whch model the lack of ft and s assumed normal dstrbuted wth mean value zero and standard devaton. The constants K and m are materal parameters. By assumng that the resduals are normal dstrbuted on a log-log scale the lkelhood functon n case of n constant ampltude tests and n 0 run-outs s gven by: n 0 nn log N log K mlog log N log K mlog Llog K, exp (3) n where N and s the number of cycles to falure and stress range for test specmen number respectvely. The parameter m s determned by least square method and the parameters log K and are estmated usng the Maxmum-Lkelhood Method where the optmzaton problem max L log K, s solved usng a standard nonlnear optmzer, e.g. the NLPQL algorthm [5]. log K, In ths paper s m assumed fxed determned by the least square method, but ths parameter could also be ncluded n the optmzaton. Snce the parameters log K and are estmated by the Maxmum-Lkelhood technque they become asymptotcally Normal dstrbuted stochastc varables wth expected values equal to the Maxmum-Lkelhood estmators and covarance equal to, see e.g. [6]: log K log K, log K Clog K, H log K, (4) log K, log K where Hlog K, s the Hessan matrx wth second order dervatves of the log-lkelhood functon. log K and denote the standard devaton on log K and respectvely. log K, s the correlaton coeffcent between log K and. The Hessan matrx s estmated by numercal dfferentaton. In table are the estmated parameters shown and the SN-curves are ftted usng all vald constant ampltude fatgue tests for the partcular R-rato and runouts are taken nto account. The parameters log K and can be assumed uncorrelated. It s noted that represents the physcal uncertanty and that and represents the statstcal uncertanty. In table are the statc log K 54

4 tenson and compresson strength gven. In fgure (left) s the constant lfe dagram contanng the SN-curves and statc strengths shown. Table : SN-curves for dfferent R-ratos for geometry R04 MD. R-rato Tests n Runouts n 0 m log K logk Table : Statc tenson strength (STT) and statc compresson strength (STC) for geometry R04 MD. Test-type Tests n Mean [MPa] Std. [MPa] STT STC amp [MPa] N=0 3 N=0 4 N=0 5 N=0 6 N=0 7 N=0 8 N=0 9 R=.0 R=0.0 R=-.5 R=-.0 R=-0.4 R=0. R= mean [MPa] Fgure. Left: Constant lfe dagram for geometry R04 MD. Rght: Lnear nterpolaton. Varable ampltude fatgue tests Varable ampltude fatgue tests are also performed wth geometry R04 MD. The load spectrum used s the wsper and wsperx spectra developed for representng the flap bendng moment of a wnd turbne blade. In order to calculate the accumulated damage D Mners rule for lnear damage accumulaton s used: n D (5) N In order to calculate the accumulated damage at falure the number of cycles to falure N has to be determned for an arbtrary R-rato. Ths s n the present paper done by lnear nterpolaton n the constant lfe dagram usng the procedure gven n the followng, see also fgure (rght). The stress cycle P s located n the constant lfe dagram Draw a lne a from the orgn through and beyond the pont P Identfy the constant lfe lnes closest to P, denoted n and n Calculate the length a on lne a between the two constant lfe lnes n and n 55

5 Calculate the length a on lne a between pont P and the constant lfe lne n Fnd the R-rato closest to P and calculate the length b between n and n ba Calculate b a Determne the stress ampltude CLD correspondng to pont Q Determne the expected number of cycles to falure N usng the SN-curve for the R-rato In table 3 are the conducted varable fatgue tests lsted together wth the mean and standard devaton on the accumulated damage at falure usng the materal parameters gven n table and. It s noted that COV D represents the model uncertanty. Table 3: Mean and standard devaton for estmated damage at falure for varable ampltude tests. Spectrum Tests N Mean Std. COV D Wsper Wsperx Reverse Wsper Reverse Wsperx Wsper, Wsperx All From table 3 t s seen that except for the wsper spectrum the estmated accumulated damage at falure s sgnfcantly below one. The uncertanty for fatgue damage accumulaton are often modelled by a lognormal dstrbuton n order two avod negatve values of Mners rule whch are physcal mpossble. The mean and standard devatons gven n table 3 can be used n a lognormal dstrbuton. Concludng remarks In the present paper are stochastc models for the uncertanty related to fatgue damage accumulaton for composte materals presented. The stochastc models are based on publc avalable constant and varable ampltude fatgue tests and the procedure used for estmatng the stochastc models are presented. Acknowledgement The work presented n ths paper s part of the project Probablstc desgn of wnd turbnes supported by the Dansh Research Agency, grant no and the project Improvement of methods for fatgue assessment supported by the Dansh Energy Authorty, EFP007 grant no The fnancal support s greatly apprecated. References [] Post NL, Case SW, Lesko JJ. Modelng the varable ampltude fatgue of composte materals: A revew and evaluaton of the state of the art for spectrum loadng. Internatonal Journal of Fatgue 008 Dec;30(): [] Mner MA. Cumulatve Damage n Fatgue. Journal of Appled Mechancs-Transactons of the Asme 945;(3):A59-A64. [3] IEC Wnd turbnes - Part: Desgn requrements. 3 rd edton [4] OPTIMAT BLADES [5] Schttkowsk K. NLPQL: A FORTRAN Subroutne Solvng Non-Lnear Programmng Problems. Annals of Operatons Research 986. [6] Lndley DV. Introducton to Probablty and Statstcs from a Baysan Vewpont. Cambrdge Unversty Press, Cambrdge;