# Initial Probability Distribution in Markov Chain Model for Fatigue Crack Growth Problem

## Authors

• Siti Sarah Januri
• Zulkifli Mohd Nopiah
• Ahmad kamal Ariffin Mohd Ihsan
• Nurulkamal Masseran
• Shahrum Abdullah

2018-09-01

## Keywords:

Initial distribution, Markov Chain model, Paris law equation, fatigue crack growth

## Abstract

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.

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