Probabilistic approach to growth and detection of a truncated distribution of initial crack lengths based on Paris' law

Abstract

A method for incorporating probabilistic considerations into fatigue life prognosis based on experimental information for Paris' law is presented in this article. A truncated probability distribution of initial crack lengths is introduced to obviate a complication of the use of Paris' law. This formulation allows the calculation of several probabilities for various values of the truncation length, including the probability of the existence of a crack larger than a predetermined critical crack length and the probability of a crack in a prescribed domain of crack lengths, and the effect of an inspection on these probabilities. A probabilistic consideration for the fact that the Paris' law parameters are not constants but distributions is also addressed using a stochastic version of Paris' law and a Monte Carlo simulation. Finally, a novel Bayesian approach which highlights the effect of probability of detection, critical crack length, and applied stress versus the number of elapsed fatigue cycles is presented and ramifications explored. This article provides insight on how various factors affect diagnosis and prognosis in a probabilistic manner.

A multi-time scale approach to remaining useful life prediction in rolling bearing

“…As the PSW is able to build a linear relationship between the tracking metric extracted from the raw sensor signal and the hidden physical damage variable, the metric is utilized as an input to a physics-based model for bearing RUL prediction. Paris law is a well-known principle that has been widely used to guide crack growth detection and prediction in mechanical materials [38][39][40]. In 1963, Paris [41] proposed an empirical crack growth model based on the stress intensity factor (SIF) as:…”

Probabilistic considerations for growth and detection of cracks from rivet holes in a lap joint

“…The experimental results reported by Virkler et al (18) indicate the parameters A and m are not constants, but distributions. Different approaches to modeling the distributions are discussed in some detail in Annis (19) and by the authors in Cohen et al (20) . In this paper, only the mean value of the distributions will be used.…”

“…For a different geometry, the authors have previously (20,21) used Paris' law to forecast to a certain number of cycles the evolution of an assumed initial distribution of cracks. If the initial distribution is expressed as f 0 (a 0 ), where a 0 is the crack size at zero cycles, the crack length distribution at N cycles, f N (a N ), can be calculated using:…”

“…The number of cycles required to reach aF for three rivet holes is NF = 53379 cycles. r a different geometry, the authors have previously (20,21) It is noted that this approach does not take into account ambiguity in crack sizing or difficulties in locating the crack, it focuses only on the detection of the crack.…”

“…This paper will consider three probabilities previously discussed in (20,21) for a surface-breaking crack in a half-space: the probability of the existence of a crack in a certain range of crack sizes, the effect of an inspection on this probability, and the probability that a crack with crack length larger than am is detected given that it exists. For these probabilities, two crack size ranges will be addressed.…”