PSI - Issue 38

Adrian Loghin et al. / Procedia Structural Integrity 38 (2022) 331–341

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A. Loghin et al. / Structural Integrity Procedia 00 (2021) 000–000

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Fig. 9: Correlated C and n data extracted from Virkler’s crack propagation measurements on 2024-T3 Aluminum alloy (Virkler (1978).

2.3. Probabilistic Assessment using Surrogate Modeling

To demonstrate the usage of the surrogate crack propagation model in a probabilistic setting, benchmark crack propagation data published in Virkler (1978) was used. This data shown as “Virkler’s data” in Figure 9a contains crack length versus cycles measurements collected from 68 constant amplitude crack growth tests under mechanical stress σ varying from 0 to 48.26 MPa at constant temperature for 2024-T3 Aluminum alloy. The specimen used for the test was M(T) with the sample width W = 152 . 4mm. In each test, the crack was grown from the initial size of a = 9mm to final size 50mm. Assuming the variation in the crack propagation data is solely due to material variation, Paris curve paired constants C , n were extracted from this data using the procedure similar to the one described in Akkaram et al. (2011). A simple crack propagation side-model was built for M(T) sample using the Paris curve da / dN = C ( ∆ K I ) n , and the K I formulation for the middle crack in a finite sheet under uniform remote tensile loading (Tada et al. (2000)): This side-model was used to determine the best fit parameters { C , n } for each crack propagation data set using the Nelder-Mead optimization routine available in SciPy optimization library (Virtanen et al. (2020)). The solid curves shown in Figure 9a are the best fit curves, and the data plotted in Figure 9b are the corresponding material { C , n } parameters from all 68 curve fits. As one may expect these two material parameters are highly correlated (Annis (2003)). Finally, the collected sample of the material parameters were used along with the normally distributed initial crack position Y 0 = N ( µ = 6.74mm, σ = 0.375mm) in Monte-Carlo simulations. Definition of Y 0 is provided in Figure 2. The crack path variation is shown in Figure 10 while the results of the RUL probabilistic assessment is given in Figure 11. Based on the observation that the Y 1 has little e ff ect on the crack growth life (Figure 7a), it can be concluded that almost all of the variation in RUL is due to the material variability in this model. The orange line plotted on top of the histogram in Figure 11b is the estimate of the probability density function obtained by Kernel Density Estimation method called Gaussian kde available in Python SciPy Library. This density function can be conveniently used for probabilistic RUL assessment. The surrogate model runtime savings are quite substantial when compared to the 3D FEA simulation. The determin istic fatigue crack growth simulation using about 220 increments is solved in under 120 minutes while surrogate model K I = σ G √ π a where G = cos ( π a / W ) − 0 . 5 (1)