Reducing Uncertainty in Fatigue Life Estimates Design, Analysis, and Simulation 1-77-NastranA Probabilistic Approach To Modeling Fatigue Life Variation Julian Raphael 1, Bart McPheeters 2, Ray DelDin 2 1. J R Technical Services, LLC, Abingdon, Virginia 24211, USA 2. NEiSoftware, Inc, Westminster, California 92683, USA Reducing Uncertainty in Fatigue Life Estimates Design, Analysis, and Simulation 1-77-NastranPresentation Outline Objective Rationale Simulation Algorithm Probabilistics Finite Element Results Fatigue Model Stochastic Results Correlated Random Variables Crack Growth Models Summary Reducing Uncertainty in Fatigue Life Estimates Design, Analysis, and Simulation 1-77-NastranObjective To develop and implement a numerical procedure that can reasonably estimate both fatigue life and fatigue life variation. Output is the Cumulative Distribution Function (CDF) that predicts life expectancy and its variation. Reducing Uncertainty in Fatigue Life Estimates Design, Analysis, and Simulation 1-77-NastranRationale For The Approach Factors of 100 in life are not uncommon for very low stress level fatigue tests Stephens, R. I., Fatemi, A., Stephens, R. R., Fuchs, H.O., Metal Fatigue in Engineering, 2nd edition, 2001 Variability in test conditions, and p, will be much smaller than the variability in material properties; all of the variability in the fatigue lives can be attributed to the material constants Socie, D., Reemsnyder, H., Downing, S., Tipton, S., et al, Fatigue Life Prediction, SAE Fatigue Design Handbook, 3rd edition, 1997 The $119 billion cost of fracture and its prevention, expressed in 1982 dollars, amounts to about 4% of the gross national product. Duga, J. J., Fisher, W. H., Buxbaum, R.W., Rosenfield, A. R., Burh, A. R., Honton, E.J., McMillan, S. C., The Economic Effects of Fracture in the United States, NBS Special Publication, 647-2, United States Department of Commerce, Washington, DC, March 1983 Reducing Uncertainty in Fatigue Life Estimates Design, Analysis, and Simulation 1-77-NastranFatigue Model Simulation Algorithm Stochastics Simulation of Material Properties Fatigue Model Damage Parameter Material Properties Data Analysis Failure CDF Stress Analysis Stress State Strain State Reducing Uncertainty in Fatigue Life Estimates Design, Analysis, and Simulation 1-77-NastranMonte Carlo Simulation Generate Random Numbers Between Zero and Unity Using Mean and Std Dev Convert the RNs to Material Constants Solve Fatigue Life Equation for Cycles to Failure, N f Analyze Failure Data to Compute CDF for Life Get Another Set of Material Constants Damage Parameter Comes From Stress And Strain States Reducing Uncertainty in Fatigue Life Estimates Design, Analysis, and Simulation 1-77-NastranProbabilistics Reducing Uncertainty in Fatigue Life Estimates Design, Analysis, and Simulation 1-77-NastranUniform Distribution Simulation N=100 Reducing Uncertainty in Fatigue Life Estimates Design, Analysis, and Simulation 1-77-NastranUniform Distribution Simulation N=1000 Reducing Uncertainty in Fatigue Life Estimates Design, Analysis, and Simulation 1-77-NastranUniform Distribution Simulation N=100000 Reducing Uncertainty in Fatigue Life Estimates Design, Analysis, and Simulation 1-77-NastranFinite Element Results Reducing Uncertainty in Fatigue Life Estimates Design, Analysis, and Simulation 1-77-NastranThe Solid Model Reducing Uncertainty in Fatigue Life Estimates Design, Analysis, and Simulation 1-77-NastranBending Stress Distribution t = 500 sec (1000 Hz Sine Wave) Units = MPa Reducing Uncertainty in Fatigue Life Estimates Design, Analysis, and Simulation 1-77-NastranLocation Of Maximum Bending Stress Crack Nucleation Is Expected To Occur At This Point. Max Principal Stress Is 1729 MPa Units = MPa Reducing Uncertainty in Fatigue Life Estimates Design, Analysis, and Simulation 1-77-NastranStress & Strain States at Expected Crack Initiation Site Reducing Uncertainty in Fatigue Life Estimates Design, Analysis, and Simulation 1-77-NastranFatemi-Socie Fatigue Model Reducing Uncertainty in Fatigue Life Estimates Design, Analysis, and Simulation 1-77-NastranDamage Parameter: Reducing Uncertainty in Fatigue Life Estimates Design, Analysis, and Simulation 1-77-NastranPlane of Maximum Damage Plane on which maximum damage occurs is not known a priori It must be calculated from the stress state, the strain state, & normal stress sensitivity The value of the damage parameter must be evaluated on every possible plane Non-proportional loading complicates the calculation Reducing Uncertainty in Fatigue Life Estimates Design, Analysis, and Simulation 1-77-NastranFatemi-Socie Crack Nucleation Plane Units = MPa Damage Parameter = Reducing Uncertainty in Fatigue Life Estimates Design, Analysis, and Simulation 1-77-NastranSimplified Fatemi-Socie Fatigue Model Reducing Uncertainty in Fatigue Life Estimates Design, Analysis, and Simulation 1-77-NastranCycles to Crack Nucleation vs Fatemi-Socie Damage Parameter f = 1758 MPa f = 2.12 b = c = G = MPa Reducing Uncertainty in Fatigue Life Estimates Design, Analysis, and Simulation 1-77-NastranStochastic Results Reducing Uncertainty in Fatigue Life Estimates Design, Analysis, and Simulation 1-77-NastranSimulation Results of Cycles to Failure With D = , E E E E E E E E E E+04 Cycles to Failure D = E E E E E E E E E E+05 Cycles to Failure D= Reducing Uncertainty in Fatigue Life Estimates Design, Analysis, and Simulation 1-77-NastranWeibull Data Analysis Linear Least Square Estimates D= N=25000 =6.744 =4499 r=0.966 F -1 (0.01)=2274 F -1 (0.10)=3222 F -1 (0.50)=4261 F -1 (0.90)=5091 D= N=25000 =3.575 =87896 r=0.959 F -1 (0.01)=24273 F -1 (0.10)=46837 F -1 (0.50)=79331 F -1 (0.90)=110991 Reducing Uncertainty in Fatigue Life Estimates Design, Analysis, and Simulation 1-77-NastranMaximum Likelihood Estimates Can Be Obtained For The Weibull Parameters Abernathy, R., The New Weibull Handbook, 4th edition, North Palm Beach, Florida, 2000 Reducing Uncertainty in Fatigue Life Estimates Design, Analysis, and Simulation 1-77-NastranSome Useful Statistical Distributions Weibull Lognormal Birnbaum-Saunders (Fatigue Life) General Extreme Value Gumbel Frechet Reducing Uncertainty in Fatigue Life Estimates Design, Analysis, and Simulation 1-77-NastranGood Statistical Fits (Compact Specimen) Lognormal Birnbaum-Saunders (Fatigue Life) Reducing Uncertainty in Fatigue Life Estimates Design, Analysis, and Simulation 1-77-NastranNot So Good Statistical Fits Both Distributions Excluded At Significance Levels Between 0.01 And 0.20 NormalWeibull Reducing Uncertainty in Fatigue Life Estimates Design, Analysis, and Simulation 1-77-NastranGoodness-of-Fit Tests Goodness-of-fit tests wont tell you what the distribution function is However, they will tell you that a candidate distribution is unsuitable at a particular significance level Some general goodness-of-fit tests Kolmogorov-Smirnov Anderson-Darling Chi Square ( ) Reducing Uncertainty in Fatigue Life Estimates Design, Analysis, and Simulation 1-77-NastranCumulative Distribution Function = = 4499 Reducing Uncertainty in Fatigue Life Estimates Design, Analysis, and Simulation 1-77-NastranProbability Density Function =6.744 =4499 Reducing Uncertainty in Fatigue Life Estimates Design, Analysis, and Simulation 1-77-NastranReliability Function (Survival Probability) =6.744 =4499 Reducing Uncertainty in Fatigue Life Estimates Design, Analysis, and Simulation 1-77-NastranHazard Function =6.744 =4499 Reducing Uncertainty in Fatigue Life Estimates Design, Analysis, and Simulation 1-77-NastranCorrelated Random Variables f and b are correlated f and c are correlated Failure to account for these correlations will overestimate actual fatigue life variation Reducing Uncertainty in Fatigue Life Estimates Design, Analysis, and Simulation 1-77-NastranOther Fatigue Models Reducing Uncertainty in Fatigue Life Estimates Design, Analysis, and Simulation 1-77-NastranCrack Propagation And Other Fatigue Models The approach is applicable to any fatigue model or crack propagation model Variations in load can be considered Variations in initial and final crack lengths can be modeled The only requirement is that the necessary CDFs be known or estimated Reducing Uncertainty in Fatigue Life Estimates Design, Analysis, and Simulation 1-77-NastranCompact Specimen Reducing Uncertainty in Fatigue Life Estimates Design, Analysis, and Simulation 1-77-NastranBasic Equations Of Crack Growth For A Compact Specimen Anderson, T. L., Fracture Mechanics: Fundamentals and Applications, 1 st edition, CRC Press, 1991 Reducing Uncertainty in Fatigue Life Estimates Design, Analysis, and Simulation 1-77-NastranSummary Monte Carlo methods are well known and appropriate for problems of this type. Solution is based on assuming the CDFs for Material Properties are Normally Distributed and known a priori. These assumptions should be replaced with experimental verification. Correlation between paired fatigue variables must be accounted for - otherwise too much variation. Reducing Uncertainty in Fatigue Life Estimates Design, Analysis, and Simulation 1-77-NastranThank You!