# Statistical Analysis Of The Virkler Data On Fatigue Crack Growth

A probabilistic framework is presented in this work for a fatigue reliability assessment considering multi-source uncertainties. These primary sources of uncertainty were considered here as coming from: (i) the inherent scattering of material properties, (ii) the variability of the main geometrical dimensions of components, (iii) the statistical nature of the actual loading conditions. In this regard, although analytical methods exist in literature (Ciavarella and Papangelo 2018), a full computational approach was adopted to develop a probabilistic FCG assessment based on a vast set of experimental data available in literature. By using the NASGRO formulation, arranged in a stochastic way, residual fatigue life predictions were computed by taking account of the main input sources reported above, see also Fig. 2. Hence, the sensitivity of each basic random variable was derived and the influences of all variables on the residual life predictions were ranked accordingly.

## Statistical Analysis Of The Virkler Data On Fatigue Crack Growth

Even though advanced probabilistic approaches can provide the engineers with more robust information about the safety of structures (Zhu et al. 2017, 2018a; Armentani et al. 2020; Niu et al. 2021; Klawonn et al. 2020; Beretta et al. 2016a; Giannella 2021a; Sepe et al. 2019; Romano et al. 2020; Giannella et al. 2018a; Grell and Laz 2010; Guida and Penta 2010; Beretta and Regazzi 2016; Beretta and Carboni 2006), as a downside, these approaches often require vast sets of experimental data for their development to limit the statistical uncertainty. The material under analysis selected for this work was the 25CrMo4 steel (known also as EA4T grade in the railway industry), for which a vast set of FCG test data at \(R=-\,1\) was available in literature (Luke et al. 2011; Náhlík et al. 2017; Pokorný et al. 2016a; Beretta et al. 2016b; Maierhofer et al. 2014; Wu et al. 2016; Varfolomeev et al. 2010; Project MARAXIL 2012; Hu et al. 2021), see Fig. 3. Chemical composition of 25CrMo4 was reported in Table 1. According to Hu et al. (2021), the matrix is primarily comprised of bainite and martensite, with an average grain diameter of around 7.2 μm; this very fine microstructure and small grain size contribute to a low level of roughness-induced crack closure (Vojtek et al. 2019; Pokorný et al. 2017).

Contrary to numerous applications (Zhu et al. 2018b, c; Guida and Penta 2010; Beretta and Regazzi 2016; Giannella et al. 2017b, 2018b, 2019a, b; Luke et al. 2011; Náhlík et al. 2017; Pokorný et al. 2016a; Beretta et al. 2016b; Giannella and Perrella 2019; Shlyannikov et al. 2021; Ayhan and Demir 2021), no numerical simulations were needed to obtain the K values along the crack-growth thanks to the simplicity of the case study considered here. Note from the Linear Elastic Fracture Mechanics (LEFM) assumption that slight discrepancies in terms of K values can return much higher deviations in terms of CGRs due to (i) the amplification through the exponent m, (ii) the scattering of material data (Zhu et al. 2018b; Giannella et al. 2018b, 2019a; Ayhan and Demir 2021; Citarella et al. 2018), see Fig. 3. A nominal initial crack \(a_\mu \) was arbitrarily defined as having a depth of 1 mm, whereas a variation of \(\pm \,0.3\) mm was envisaged through a standard deviation \(a_\sigma \) of 0.15 (assuming a \(\mu \pm 2\sigma \) range). The geometrical tolerance for the diameter was approximately taken as \(D_\sigma =0.015D_\mu \) of variation from the designed nominal size \(D_\mu \) equal to 18 mm. The corresponding variation was therefore set up to \(18 \pm 0.5\) mm if considering that a range of \(\mu \pm 2\sigma \) covers nearly 95.5% of a normal distribution.

Crack propagation phenomena are inherently stochastic and the solutions obtained deterministically are only limiting outcomes that do not allow for their comprehensive description. The stochastic procedure for the fatigue life assessment of the round bar was performed within a user-made MATLAB (MATLAB 2019) routine implementing the logic illustrated in Figs. 5 and 6. The routine considered as input data the variables presented in the previous Sects. 2 and 3 and provided as output the predictions of the residual fatigue lives estimated for the cracked round bar.