Initial Probability Distribution in Markov Chain Model for Fatigue Crack Growth Problem
Keywords:Initial distribution, Markov Chain model, Paris law equation, fatigue crack growth
Stochastic processes in fatigue crack growth problem usually due to the uncertainties factors such as material properties, environmental conditions and geometry of the component. These random factors give an appropriate framework for modelling and predicting a lifetime of the structure. In this paper, an approach of calculating the initial probability distribution is introduced based on the statistical distribution of initial crack length. The fatigue crack growth is modelled and the probability distribution of the damage state is predicted by a Markov Chain model associated with a classical deterministic crack Paris law. It has been used in calculating the transition probabilities matrix to represent the physical meaning of fatigue crack growth problem. The initial distribution has been determined as lognormal distribution which 66% that the initial crack length will happen in the first state. The data from the experimental work under constant amplitude loading has been analyzed using the Markov Chain model. The results show that transition probability matrix affect the result of the probability distribution and the main advantage of the Markov Chain is once all the parameters are determined, the probability distribution can be generated at any time, x.
 Drewniak, J., & Hojdys, L. (2015). The Method of Analysis of Fatigue Crack Growth by Bogdanow-Kozin Model. Machine Dynamics Research, 39(4), 125â€“132.
 Ellyin, F. (1997). Fatigue Damage , Crack Growth and Life Prediction. Chapman & Hall, 2-6 Boundary Row, London SE18HN, UK.
 Gansted, L., Brincker, R., & Hansen, L. P. (1991). Fracture mechanical Markov chain crack growth model. Engineering Fracture Mechanics, 38(6), 475â€“489. http://doi.org/10.1016/0013-7944(91)90097-K
 Khelif, R., Chateauneuf, A., & Chaoui, K. (2008). Statistical analysis of HDPE fatigue lifetime. Meccanica, 43(April), 567â€“576. http://doi.org/10.1007/s11012-008-9133-7
 KocaÅ„da, D., & Jasztal, M. (2012). Probabilistic predicting the fatigue crack growth under variable amplitude loading. International Journal of Fatigue, 39, 68â€“74. http://doi.org/10.1016/j.ijfatigue.2011.03.011
 Kozin, F., & Bogdanoff, J. (1981). A Critical Analysis of Some Probabilistic Models for Fatigue Crack Growth. Engineering Fracture Mechanics, 14(MC), 59â€“89.
 Kozin, F., & Bogdanoff, J. L. (1983). On the probabilistic modeling of fatigue crack growth. Engineering Fracture Mechanics, 18(3), 623â€“632. http://doi.org/10.1016/0013-7944(83)90055-3
 Lee, O. S. (1998). Fatigue Life Prediction by Statistical Approach Under Constant Amplitude Loading. KSME International Journal, 12(1).
 Makkonen, M. (2009). Predicting the total fatigue life in metals. International Journal of Fatigue, 31(7), 1163â€“1175. http://doi.org/10.1016/j.ijfatigue.2008.12.008
 Ossai, C. I., Boswell, B., & Davies, I. (2016). Markov chain modelling for time evolution of internal pitting corrosion distribution of oil and gas pipelines. Engineering Failure Analysis, 60, 209â€“228. http://doi.org/10.1016/j.engfailanal.2015.11.052
 Rastogi, R., Ghosh, S., Ghosh, A. K., Vaze, K. K., & Singh, P. K. (2016). Fatigue crack growth prediction in nuclear piping using Markov chain Monte Carlo simulation. Fatigue & Fracture of Engineering Materials & Structures. http://doi.org/10.1111/ffe.12486
 Santecchia, E., Hamouda, A. M. S., Musharavati, F., Zalnezhad, E., Cabibbo, M., Mehtedi, M. El, & Spigarelli, S. (2016). A Review on Fatigue Life Prediction Methods for Metals. Advances in Materials Science and Engineering Volume, 2016. http://doi.org/10.1155/2016/9573524
 Son, K. S., Kim, D. H., Kim, C. H., & Kang, H. G. (2016). Study on the systematic approach of Markov modeling for dependability analysis of complex fault-tolerant features with voting logics. Reliability Engineering & System Safety, 150, 44â€“57. http://doi.org/10.1016/j.ress.2016.01.014
 Wu, W. F., & Ni, C. C. (2003). A study of stochastic fatigue crack growth modeling through experimental data. Probabilistic Engineering Mechanics, 18(2), 107â€“118. http://doi.org/10.1016/S0266-8920(02)00053-X
 Wu, W. F., & Ni, C. C. (2004). Probabilistic models of fatigue crack propagation and their experimental verification. Probabilistic Engineering Mechanics, 19, 247â€“257. http://doi.org/10.1016/j.probengmech.2004.02.008