Sørensen, John Dalsgaard

Toft, Henrik Stensgaard

English

In the present paper stochastic models for fatigue damage accumulation for composite materials are presented based on public available constant and variable amplitude fatigue tests. The methods used for estimating the SN-curve and accumulated fatigue damage are presented.In the present paper stochastic models for fatigue damage accumulation for composite materials are presented based on public available constant and variable amplitude fatigue tests. The methods used for estimating the SN-curve and accumulated fatigue damage are presented

    Aalborg UniversitetUncertainty on Fatigue Damage Accumulation for Composite MaterialsToft, Henrik Stensgaard; Sørensen, John DalsgaardPublished in:Proceedings of the Twenty Second Nordic Seminar on Computational MechanicsPublication date:2009Document VersionPublisher's PDF, also known as Version of recordLink to publication from Aalborg UniversityCitation for published version (APA):Toft, H. S., & Sørensen, J. D. (2009). Uncertainty on Fatigue Damage Accumulation for Composite Materials. InL. Damkilde, L. Andersen, A. S. Kristensen, & E. Lund (Eds.), Proceedings of the Twenty Second NordicSeminar on Computational Mechanics (pp. 253-256). Department of Civil Engineering, Aalborg University. 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You may freely distribute the URL identifying the publication in the public portal ?Take down policyIf you believe that this document breaches copyright please contact us at vbn@aub.aau.dk providing details, and we will remove access tothe work immediately and investigate your claim.Downloaded from vbn.aau.dk on: August 22, 2021Uncertainty on Fatigue Damage Accumulation for Composite Materials Henrik Stensgaard Toft* and John Dalsgaard SørensenDepartment of Civil Engineering Aalborg University, Aalborg, Denmark  email: hst@civil.aau.dk Summary In the present paper stochastic models for fatigue damage accumulation for composite materials are presented based on public available constant and variable amplitude fatigue tests. The methods used for estimating the SN-curve and accumulated fatigue damage are presented. IntroductionDamage accumulation models for composite materials exposed to fatigue loading have been widely considered in the literature, see e.g. [1] for a review. However, even though new empirical and physical models for the accumulation of damage are proposed, these models do not seem to perform much better than the linear damage accumulation proposed by Miner [2]. For this reason the accumulated damage is normally determined by Miners rule as recommended in [3] for wind turbine blades. The uncertainties in damage accumulation based on Miners rule can in general be divided into three parts:  Physical uncertainty on the SN-curves  Statistical uncertainty on the SN-curves due to a limited number of tests  Model uncertainty on Miners rule The physical uncertainty on the SN-curves is due to the natural inherent uncertainty in the material and can not be reduced. The statistical uncertainty can be reduced by performing additional fatigue tests and the model uncertainty can in principle be reduced by adopting a better model. In the present paper is the physical and statistical uncertainty on SN-curves for composite material determined based on a number of constant amplitude fatigue tests performed with different mean stresses. Based on variable amplitude fatigue tests for the same material using a standard load spectrum and the estimated SN-curves based on constant amplitude fatigue tests is the model uncertainty on Miners rule determined. In most standards and regulations including [3] are fatigue design performed by using a deterministic design approach. However, calculation of the accumulated fatigue damage includes significant uncertainties for which reason a probabilistic design approach should be adopted in order to take the individual uncertainties into account in a rational manner. The stochastic models determined in the present paper forms the basis for a probabilistic modeling of the considered class of composite materials in fatigue loading. Constant amplitude fatigue tests The constant amplitude and variable amplitude fatigue tests used in the present paper are given in the OptiDAT database [4] for geometry R04 MD (MultiDirectional laminate). This geometry has been selected due to the many fatigue tests performed with this geometry. For composite materials the mean stress can have a significant influence on the fatigue properties which can be taken into 253Proceedings of the Twenty Second Nordic Seminar on Computational Mechanics© Aalborg University 2009 | ISSN 1901-7278 DCE Technical Memorandum No. 11account by calculation of an SN-curve for different R-ratios and arranging these in a constant life diagram. The R-ratio is defined by: minmaxR ~~ (1)where min  and max  are the minimum and maximum stress in a stress cycle respectively. Different types of SN-curves have been used for composite material, but no specific SN-curve have been recommended in [3]. In the present study is a log-log SN-curve used: log log logN K m      (2)where N  is the number of cycles to failure,   is the stress range and   is parameter which model the lack of fit and is assumed normal distributed with mean value zero and standard deviation  . The constants K  and m  are material parameters. By assuming that the residuals are normal distributed on a log-log scale the likelihood function in case of n  constant amplitude tests and 0n  run-outs is given by: 	  	  	 021 1log log log log log log1 1log , exp22n nni i i ii i nN K m N K mL K   ~~                          (3)where iN  and i  is the number of cycles to failure and stress range for test specimen number irespectively. The parameter m  is determined by least square method and the parameters log Kand   are estimated using the Maximum-Likelihood Method where the optimization problem 	 log ,max log ,KL K~~ is solved using a standard nonlinear optimizer, e.g. the NLPQL algorithm [5]. In this paper is m  assumed fixed determined by the least square method, but this parameter could also be included in the optimization. Since the parameters log K  and   are estimated by the Maximum-Likelihood technique they become asymptotically Normal distributed stochastic variables with expected values equal to the Maximum-Likelihood estimators and covariance equal to, see e.g. [6]: 21 log log , loglog , log , 2log , logK K KK KK KC H      ~~ ~~  ~~                  (4) where log ,KH   is the Hessian matrix with second order derivatives of the log-Likelihood function. log K  and ~~  denote the standard deviation on log K  and   respectively. log ,K   is the correlation coefficient between log K  and  . The Hessian matrix is estimated by numerical differentiation.In table 1 are the estimated parameters shown and the SN-curves are fitted using all valid constant amplitude fatigue tests for the particular R-ratio and runouts are taken into account. The parameters log K  and   can be assumed uncorrelated. It is noted that   represents the physical uncertainty and that log K  and ~~  represents the statistical uncertainty. In table 2 are the static 254tension and compression strength given. In figure 1 (left) is the constant life diagram containing the SN-curves and static strengths shown. Table 1: SN-curves for different R-ratios for geometry R04 MD. R-ratio Tests n Runouts n0 m log K  logK 0.5 15 0 10.541 27.768 0.358 0.092 0.065 0.1 45 2 9.508 27.191 0.259 0.039 0.027 -0.4 28 0 7.582 23.398 0.435 0.082 0.058 -1.0 84 3 6.719 21.359 0.878 0.094 0.068 -2.5 10 2 11.983 35.231 0.633 0.197 0.143 10.0 34 0 22.211 58.664 0.644 0.110 0.078 2.0 6 3 29.686 73.780 0.354 0.143 0.103 Table 2: Static tension strength (STT) and static compression strength (STC) for geometry R04 MD. Test-type Tests n Mean [MPa] Std. [MPa] STT 66 556.5 64.2 STC 55 -458.6 33.2 -600 -400 -200 0 200 400 6000100200300mean [MPa] amp [MPa]R=0.5R=0.1R=-0.4R=-1.0R=-2.5R=10.0R=2.0N=103N=104N=105N=106N=107N=108N=109                Figure 1. Left: Constant life diagram for geometry R04 MD. Right: Linear interpolation. Variable amplitude fatigue tests Variable amplitude fatigue tests are also performed with geometry R04 MD. The load spectrum used is the wisper and wisperx spectra developed for representing the flap bending moment of a wind turbine blade. In order to calculate the accumulated damage D  Miners rule for linear damage accumulation is used: 	 11ni iDN  (5)In order to calculate the accumulated damage at failure the number of cycles to failure N  has to be determined for an arbitrary R-ratio. This is in the present paper done by linear interpolation in the constant life diagram using the procedure given in the following, see also figure 1 (right).  The stress cycle P  is located in the constant life diagram  Draw a line a  from the origin through and beyond the point P Identify the constant life lines closest to P , denoted 1n  and 2n Calculate the length 1a  on line a  between the two constant life lines 1n  and 2n255 Calculate the length 2a  on line a  between point P  and the constant life line 2n Find the R-ratio closest to P  and calculate the length 1b  between 1n  and 2n Calculate 1 221b aba Determine the stress amplitude CLD  corresponding to point Q Determine the expected number of cycles to failure N  using the SN-curve for the R-ratio In table 3 are the conducted variable fatigue tests listed together with the mean and standard deviation on the accumulated damage at failure using the material parameters given in table 1 and 2. It is noted that COVD represents the model uncertainty. Table 3: Mean and standard deviation for estimated damage at failure for variable amplitude tests. Spectrum Tests N Mean  Std. COVDWisper 10 0.9000 0.5355 0.595 Wisperx 13 0.2763 0.1982 0.717 Reverse Wisper 2 0.2020 - - Reverse Wisperx 10 0.3179 0.1576 0.496 Wisper, Wisperx 23 0.5474 0.4886 0.893 All 35 0.4621 0.4196 0.908 From table 3 it is seen that except for the wisper spectrum the estimated accumulated damage at failure is significantly below one. The uncertainty for fatigue damage accumulation are often modelled by a lognormal distribution in order two avoid negative values of Miners rule which are physical impossible. The mean and standard deviations given in table 3 can be used in a lognormal distribution. Concluding remarks In the present paper are stochastic models for the uncertainty related to fatigue damage accumulation for composite materials presented. The stochastic models are based on public available constant and variable amplitude fatigue tests and the procedure used for estimating the stochastic models are presented. Acknowledgement The work presented in this paper is part of the project “Probabilistic design of wind turbines” supported by the Danish Research Agency, grant no. 2104-05-0075 and the project “Improvement of methods for fatigue assessment” supported by the Danish Energy Authority, EFP2007 grant no. 33033-0077. The financial support is greatly appreciated. References [1]  Post NL, Case SW, Lesko JJ. Modeling the variable amplitude fatigue of composite materials: A review and evaluation of the state of the art for spectrum loading. International Journal of Fatigue 2008 Dec;30(12):2064-86. [2]  Miner MA. Cumulative Damage in Fatigue. Journal of Applied Mechanics-Transactions of the Asme 1945;12(3):A159-A164. [3]  IEC 61400-1. Wind turbines - Part1: Design requirements. 3rd edition. 2005.  [4]  OPTIMAT BLADES.  2006.  www.kc-wmc.nl.[5]  Schittkowski K. NLPQL: A FORTRAN Subroutine Solving Non-Linear Programming Problems. Annals of Operations Research 1986. [6]  Lindley DV. Introduction to Probability and Statistics from a Baysian Viewpoint. Cambridge University Press, Cambridge; 1976.256

Uncertainty on Fatigue Damage Accumulation for Composite Materials

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Uncertainty on Fatigue Damage Accumulation for Composite Materials

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