A new probabilistic model for crack propagation under fatigue loads and its connection with Wöhler fields

This paper deals with the problem of building crack growth models and connecting them with Wohler field models. After, analyzing the physical validity of some crack growth models, a general methodology to build models is presented starting by identifying the set of variables involved in the crack growth problem. Next, a minimum subset of dimensionless parameters using the Buckingham Π theorem is selected. The next step consists of imposing some consistency and compatibility conditions in terms of functional equations, which, once solved, provide the subset of crack growth models satisfying the required deterministic and stochastic compatibility conditions, which, in addition to providing mean values of the crack sizes as a function of time, as alternative models do, also give densities of the crack sizes. As a result, the main elements required to build a crack growth model, such as the initial crack size distribution, the crack growth function and a loading effect functions, have been identified. Finally, the compatibility of Wohler field and crack growth models is dealt with and the advantages of such a dual treatment is emphasized. The methodology is illustrated with some examples, including crack growth for different load histories, and some conclusions are given.

[1]  Enrique Castillo,et al.  Lifetime regression models based on a functional equation of physical nature , 1987, Journal of Applied Probability.

[2]  E. C. Ron,et al.  Functional equations and modelling in science and engineering , 1992 .

[3]  Ali S. Hadi,et al.  Extreme Value and Related Models with Applications in Engineering and Science , 2004 .

[4]  R. Forman,et al.  Numerical Analysis of Crack Propagation in Cyclic-Loaded Structures , 1967 .

[5]  V. V. Bolotin,et al.  Probabilistic model of early fatigue crack growth , 1998 .

[6]  Chantier,et al.  A probabilistic approach to predict the very high-cycle fatigue behaviour of spheroidal graphite cast iron structures , 2000 .

[7]  V. V. Bolotin,et al.  Early fatigue crack growth as the damage accumulation process , 2001 .

[8]  G. Deierlein,et al.  Cyclic Void Growth Model to Assess Ductile Fracture Initiation in Structural Steels due to Ultra Low Cycle Fatigue , 2007 .

[9]  F. Erdogan,et al.  Fatigue and fracture of cylindrical shells containing a circumferential crack , 1970 .

[10]  J. Kohout,et al.  A new function describing fatigue crack growth curves , 1999 .

[11]  W. Dixon.  The Up-and-Down Method for Small Samples , 1965 .

[12]  Wen-Fang Wu,et al.  A study of stochastic fatigue crack growth modeling through experimental data , 2003 .

[13]  M. Klesnil,et al.  Effect of stress cycle asymmetry on fatigue crack growth , 1972 .

[14]  A. Fernández‐Canteli,et al.  A statistical model for crack growth based on tension and compression Wöhler fields , 2008 .

[15]  Enrique Castillo.  Extreme value theory in engineering , 1988 .

[16]  J. Holmén.  Fatigue of Concrete by Constant and Variable Amplitude loading , 1982 .

[17]  Bernard D. Coleman,et al.  Statistics and Time Dependence of Mechanical Breakdown in Fibers , 1958 .

[18]  Fa Bastenaire,et al.  NEW METHOD FOR THE STATISTICAL EVALUATION OF CONSTANT STRESS AMPLITUDE FATIGUE-TEST RESULTS , 1971 .

[19]  François Hild,et al.  HIGH-CYCLE FATIGUE BEHAVIOUR OF SPHEROIDAL GRAPHITE CAST IRON , 1998 .

[20]  L. Faravelli,et al.  Inherent variability of an experimental crack growth curve , 2007 .

[21]  H. Saunders,et al.  Probabilistic models of cumulative damage , 1985 .

[22]  Hirokazu Shoji,et al.  A Bayesian evaluation of simulation models of multiple-site fatigue cracks , 2001 .

[23]  J. Aczél,et al.  Lectures on Functional Equations and Their Applications , 1968 .

[24]  D. V. Ramsamooj,et al.  Analytical Prediction of Fatigue Crack Propagation in Metals , 2003 .

[25]  Ali S. Hadi,et al.  On fitting a fatigue model to data , 1999 .

[26]  W. Ames.  Mathematics in Science and Engineering , 1999 .

[27]  Alfonso Fernández-Canteli,et al.  A general regression model for lifetime evaluation and prediction , 2001 .

[28]  Marios K. Chryssanthopoulos,et al.  Fatigue and fracture simulation of welded bridge details through a bi-linear crack growth law , 2004 .

[29]  L. Molent,et al.  Recent developments in fatigue crack growth assessment , 2006 .

[30]  Enrique Castillo,et al.  A Parametric Lifetime Model for the Prediction of High‐Cycle Fatigue Based on Stress Level and Amplitude , 2006 .

[31]  R. Ritchie.  Incomplete self-similarity and fatigue-crack growth , 2005 .

[32]  Andrea Spagnoli,et al.  Self-similarity and fractals in the Paris range of fatigue crack growth , 2005 .

[33]  Bruce R. Ellingwood,et al.  Evaluation of crack growth in miter gate weldments using stochastic fracture mechanics , 2001 .

[34]  Alfonso Fernández-Canteli,et al.  A general model for fatigue damage due to any stress history , 2008 .

[35]  E. Siores,et al.  Recent Developments in Fatigue Crack Growth , 2006 .

[36]  Alexander M. Mood,et al.  A Method for Obtaining and Analyzing Sensitivity Data , 1948 .

[37]  J N Yang,et al.  Stochastic Crack Propagation with Applications to Durability and Damage Tolerance Analyses , 1985 .

[38]  Wen-Fang Wu,et al.  Probabilistic models of fatigue crack propagation and their experimental verification , 2004 .

[39]  Enrique Castillo,et al.  Functional Equations in Applied Sciences , 2004 .

[40]  P. C. Paris,et al.  A Critical Analysis of Crack Propagation Laws , 1963 .

[41]  Robert E. Melchers,et al.  Overload failure of structural components under random crack propagation and loading - a random process approach , 2004 .

[42]  V. V. Bolotin.  Mechanics of Fatigue , 1999 .

[43]  R N Hill,et al.  Comparison of the up-and-down, conventional LD50, and fixed-dose acute toxicity procedures. , 1995, Food and chemical toxicology : an international journal published for the British Industrial Biological Research Association.

[44]  Ali S. Hadi,et al.  A fatigue model with local sensitivity analysis , 2007 .

[45]  Andrea Carpinteri,et al.  A fractal analysis of size effect on fatigue crack growth , 2004 .