First Passage Solutions in Fatigue Crack Propagation

Abstract

THE STATISTICAL CHARACTERISTICS OF THE TIME REQUIRED BY THE CRACK SIZE TO REACH A SPECIFIED LENGTH ARE SOUGHT. THIS TIME IS TREATED AS THE RANDOM VARIABLE TIME-TO-FAILURE AND THE ANALYSIS IS CAST INTO A FIRST-PASSAGE TIME PROBLEM. THE FATIGUE CRACK PROPAGATION GROWTH EQUATION IS RANDOMIZED BY EMPLOYING THE PULSE TRAIN STOCHASTIC PROCESS MODEL. THE RESULTING EQUATION IS STOCHASTICALLY AVERAGED SO THAT THE CRACK SIZE CAN BE APPROX. MODELED AS MARKOV PROCESS. CHOOSING THE APPROPRIATE TRANSITION DENSITY FUNCTION FOR THIS PROCESS AND SETTING THE PROPER INITIAL AND BOUNDARY CONDITIONS IT BECOMES POSSIBLE TO SOLVE THE ASSOCIATED FORWARD KOLMOGOROV EQUATION EXPRESSING THE SOLUTION IN THE FORM OF AN INFINITE SERIES. NEXT, THE SURVIVAL PROBABILITY OF A COMPONENT, THE CUMULATIVE DISTR. FUNCTION AND THE PROBABILITY DENSITY FUNCTION OF THE FIRST PASSAGE TIME ARE DETERMINED IN A SERIES FORM AS WELL. CORRESPONDING EXPRESSIONS ARE ALSO DERIVED FOR ITS MEAN AND MEAN SQUARE. VERIFICATION OF THE THEORETICAL RESULTS IS ATTEMPTED THROUGH COMPARISONS WITH ACTUAL EXPERIMENTAL DATA AND NUMERICAL SIMULATION STUDIES.