Reducing Uncertainty in Fatigue Life Estimates Design, Analysis, and Simulation 1-77-Nastran A 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-Nastran Presentation 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-Nastran Objective 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-Nastran Rationale 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-Nastran Fatigue 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-Nastran Monte 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-Nastran Probabilistics

Reducing Uncertainty in Fatigue Life Estimates Design, Analysis, and Simulation 1-77-Nastran Uniform Distribution Simulation N=100

Reducing Uncertainty in Fatigue Life Estimates Design, Analysis, and Simulation 1-77-Nastran Uniform Distribution Simulation N=1000

Reducing Uncertainty in Fatigue Life Estimates Design, Analysis, and Simulation 1-77-Nastran Uniform Distribution Simulation N=100000

Reducing Uncertainty in Fatigue Life Estimates Design, Analysis, and Simulation 1-77-Nastran Finite Element Results

Reducing Uncertainty in Fatigue Life Estimates Design, Analysis, and Simulation 1-77-Nastran The Solid Model

Reducing Uncertainty in Fatigue Life Estimates Design, Analysis, and Simulation 1-77-Nastran Bending Stress Distribution t = 500  sec (1000 Hz Sine Wave) Units = MPa

Reducing Uncertainty in Fatigue Life Estimates Design, Analysis, and Simulation 1-77-Nastran Location 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-Nastran Stress & Strain States at Expected Crack Initiation Site

Reducing Uncertainty in Fatigue Life Estimates Design, Analysis, and Simulation 1-77-Nastran Fatemi-Socie Fatigue Model

Reducing Uncertainty in Fatigue Life Estimates Design, Analysis, and Simulation 1-77-Nastran Damage Parameter:

Reducing Uncertainty in Fatigue Life Estimates Design, Analysis, and Simulation 1-77-Nastran Plane 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-Nastran Fatemi-Socie Crack Nucleation Plane Units = MPa Damage Parameter =

Reducing Uncertainty in Fatigue Life Estimates Design, Analysis, and Simulation 1-77-Nastran Simplified Fatemi-Socie Fatigue Model

Reducing Uncertainty in Fatigue Life Estimates Design, Analysis, and Simulation 1-77-Nastran Cycles 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-Nastran Stochastic Results

Reducing Uncertainty in Fatigue Life Estimates Design, Analysis, and Simulation 1-77-Nastran Simulation 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-Nastran Weibull 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-Nastran Maximum 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-Nastran Some 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-Nastran Good Statistical Fits (Compact Specimen) Lognormal Birnbaum-Saunders (Fatigue Life)

Reducing Uncertainty in Fatigue Life Estimates Design, Analysis, and Simulation 1-77-Nastran Not 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-Nastran Goodness-of-Fit Tests Goodness-of-fit tests won’t 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-Nastran Cumulative Distribution Function  =  = 4499

Reducing Uncertainty in Fatigue Life Estimates Design, Analysis, and Simulation 1-77-Nastran Probability Density Function  =6.744  =4499

Reducing Uncertainty in Fatigue Life Estimates Design, Analysis, and Simulation 1-77-Nastran Reliability Function (Survival Probability)  =6.744  =4499

Reducing Uncertainty in Fatigue Life Estimates Design, Analysis, and Simulation 1-77-Nastran Hazard Function  =6.744  =4499

Reducing Uncertainty in Fatigue Life Estimates Design, Analysis, and Simulation 1-77-Nastran Correlated 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-Nastran Other Fatigue Models

Reducing Uncertainty in Fatigue Life Estimates Design, Analysis, and Simulation 1-77-Nastran Crack 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-Nastran Compact Specimen

Reducing Uncertainty in Fatigue Life Estimates Design, Analysis, and Simulation 1-77-Nastran Basic 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-Nastran Summary 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-Nastran Thank You!