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When the fatigue occurs above 103 cycles (usually 104 or more), it is usually called High-cycle fatigue.
The S-N curve for a specific material is the curve of nominal stress S (y axis) against the number of cycles to failure N (x axis).
The S-N curve of 1045 steel and 2014-T6 aluminum alloy is enclosed below to represent two tipical S-N curves of metal materials. In short, the S-N curve is used to predicts the number of cycles sustained under certain stress before failure. While S-N curve is clear and straight forward on addressing the service life under fatigue, its accuracy leaves some room to be improved. The non-zero mean stress S-N relation requires huge amount of experiments to obtain the required data and form the mesh over a wide range of mean stresses. A more popular diagram for design purposes is called master diagram which accumulates fatigue data under different mean stresses and presents each line as the fatigue life under the net of maximum and minimum stresses in addition to mean stress and alternating stress as the reference axises.
Please note that the S-N curve and its elaborated master diagram require a lot of experiments to accumulate the necessary data. With the complexity of a master diagram, not to mention the time and effort to create one, the fatigue prediction is still less that perfect. Palmgren (1924) and Miner (1945) suggested an algorithm to combine individual contributions, known as Palmgren-Miner's linear damage hypothesis or Miner's rule. Miner's rule assume the fatigue life is consumed by the linear combination of different portion of stress state, both cycles and magnitude. Based on the crack opening displacement, fatigue crack growth model is expressed as a function of mechanical properties.
All structural component experiences cyclic stresses of sufficient magnitude during their life span. Equation (5) and (6) are solved with the initial parameters as given by Virkler et al [3] and model predictions are compared in Fig 1.
The stochastic crack growth model based on crack opening and random process parameter for the effect of material non-homogeneity is found to be suitable for crack growth prediction. Two components with exactly the same alloy can have vastly different mechanical propoerties.

When the amplitude of repeat loading is below the fatigue limit, small stresses do not shorten the fatigue life of the material.
The curve gives designers a quick reference of the allowable stress level for an intended service life. Partially because of the statistical nature of fatigue and Partially because of the difference between laboratory experiments and the real-life practice.
However, it is more often the time-varying stresses are oscillating near a non-zero mean stress.
On the other hand, Goodman and Gerber's approximations, although simple, they might not properly represent the specific material. In practice, a mechanical component is exposed to a complex, often random, sequence of loads, large and small and different mean values. This approximation, which is simple and straight forward, does not take the squences of loading history into account.
The effect of material non-homogeneity is included in the model through a random process parameter of Gaussian type.
Under fluctuating load condition the physical mechanisms leading to crack formation and failure is very complicated and difficult to model. The predicted a-N curve seems to be very close to the experimental results of Virkler data. 2 Prediction of mean a – N curve from the four generated sample curves at 90% probability and 95% confidence level within an error of 5%. 2 shows the predicted a - N curve from the four generated sample curves at 90% probability and 95% confidence level within an error of 5%. For example, a serial of high stress loading, which weaken the material, followed by a serial low stress loading may cause more damage than a serial of low stress loading followed by a serial of high stress loading. The model is validated through the experimental and predicted results from several data sets.
The a – N curve thus obtained explains only the variability due to the material properties and specimen geometry. The random factor X (a) now is added to the each normalized life data (normalized between -1 and 1) obtained for each crack increment from Eq.

It is widely recognized that the fatigue crack growth is fundamentally a stochastic phenomenon. Hence, the variability due to material non-homogeneity is to be incorporated to account this effect on the growth process.
The two main reasons for the randomness in fatigue crack growth behaviour are the random material resistance or inhomogeneous material properties and the random loading.
4-6 shows the predicted mean S-N curve along with the experimental data for different materials.
During the last three decades, the probabilistic aspect of fatigue crack growth has been addressed by many researchers [1-7].
The results shown in figures are very close to the experimental results which show the capability of the model presented in this study. These studies are based on Markov chain model, random process model or random variable model. Most of the stochastic crack growth models are based on the inclusion of a suitable stochastic random process either of stationary and ergodic process of Gaussian type or non Gaussian type in the deterministic crack growth model. Mostly the deterministic part of the model is based on Paris-Erdogan, Elber or polynomial models. Keeping in view these aspects, the present study is aimed at the development of a stochastic crack growth model and estimation of confidence and probability bounded crack growth relation (a-N) curve) coupled with stationary Gaussian random process. In the model the statistical scatter of the material properties between the specimens and the microstructural stochastic non-homogeneity of the material within a specimen are well addressed. The validity of the model is demonstrated through the comparison with an extensive amount of the published crack growth data. The probability-confidence bounded prediction of a-N curves presented in this paper will be extremely helpful for the reliability assessment of structure.

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