Fracture Mechanics

Before proceeding it is worth recalling a few concepts of Fracture Mechanics, which shall turn out to be useful later on.

Stress Intensity Factor Range

It is well-established that the severity of fatigue loads in the neighbouring region of a crack can be assess via the stress intensity factor range (SIF) \(\Delta K\):

\[\Delta K = Y\, \Delta\sigma \sqrt{\pi\, a}\]

where \(\Delta\sigma\) is the applied stress range, \(Y\) is the geometric factor of the crack, and \(a\) is the characteristic length of the crack.

El-Haddad Curve

Assuming \(\Delta K\) of the prior section as the fatigue crack driving force, it is possible to outline the fatigue endurance limit of flawed or cracked metals exploiting the El-Haddad (EH) curve, which is a semi-empirical model based on Linear Fracture Mechanics.

For the sake of illustrating the package’s functionalities, we consider a metallic alloy containing defects. Furthermore, we restate \(\Delta K\), by associating a characteristic crack length to the defects, in agreement with Murakami [1]. Hence:

\[a\mapsto \sqrt{\text{area}}\]

where \(\sqrt{\text{area}}\) is the projected area of the defect onto the plane orthogonal to the direction of the applied fatigue load. Accordingly, the SIF range turns out to be:

\[\Delta K = Y\, \Delta\sigma \sqrt{\pi\, \sqrt{\text{area}}}\]

where, in this case, \(Y\) is determined upon the distance between the defect’s centroid and the free surface of the fatigue specimen [2].

Given the above, the EH curve is analytically defined as [3]:

\[\Delta\sigma = \Delta\sigma_w \sqrt{{{\sqrt{\text{area}}}\over{\sqrt{\text{area}_0}+\sqrt{\text{area}}}}}\]

where \(\Delta\sigma_w\) is the fatigue endurance limit of the defect-free specimen, and \(\sqrt{\text{area}_0}\) is the EH critical length, defined using the inverse of the SIF range for defects:

\[\sqrt{\text{area}_0} = \frac{1}{\pi} \bigg(\frac{\Delta K_{th,lc}}{Y\,\Delta\sigma_w}\bigg)^2\]

in which \(\Delta K_{th,lc}\) is the SIF range threshold for long cracks. It is therefore clear that both \(\Delta K_{th,lc}\) and \(\Delta\sigma_w\) must be determined if one wishes to estimate the fatigue endurance limit of a given material. To do so, B-FADE implements the Maximum a Posteriori Estimation (MAP) approach showcased in [4]. Once the posterior distribution of the parameters is known, it is opportunely post-processed the outline the probabilistic propagating crack region via both frequentist and Bayesian approaches.

References

[1]

Yukitaka Murakami. Metal Fatigue: Effects of Small Defects and Nonmetallic Inclusions. Elsevier, 2019.

[2]

Enrico Salvati, Alessandro Tognan, Luca Laurenti, Marco Pelegatti, and Francesco De Bona. A defect-based physics-informed machine learning framework for fatigue finite life prediction in additive manufacturing. Materials & Design, 222:111089, October 2022. doi:10.1016/j.matdes.2022.111089.

[3]

Uwe Zerbst, Giovanni Bruno, Jean-Yves Buffière, Thomas Wegener, Thomas Niendorf, Tao Wu, Xiang Zhang, Nikolai Kashaev, Giovanni Meneghetti, Nik Hrabe, Mauro Madia, Tiago Werner, Kai Hilgenberg, Martina Koukolíková, Radek Procházka, Jan Džugan, Benjamin Möller, Stefano Beretta, Alexander Evans, Rainer Wagener, and Kai Schnabel. Damage tolerant design of additively manufactured metallic components subjected to cyclic loading: State of the art and challenges. Progress in Materials Science, 121:100786, August 2021. doi:10.1016/j.pmatsci.2021.100786.

[4]

Alessandro Tognan and Enrico Salvati. Probabilistic defect-based modelling of fatigue strength for incomplete datasets assisted by literature data. International Journal of Fatigue, 173:107665, August 2023. doi:10.1016/j.ijfatigue.2023.107665.