PSI - Issue 79

João Alves et al. / Procedia Structural Integrity 79 (2026) 326–334

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4. Simulation Procedure For this research, a new SN predicted model based on equation 14 is proposed through the integration of equation 6. The variable, a i , assumes the total projected area of the initial defects obtained in nanotomography, and the final defect or the maximum allowable defect, a f , will assume a value of 1 mm as proposed by (Murakami & Endo, 2023), and a value of 2 mm for comparing the influence of the assumed final defect. σ = σ ×(1+( ( ( × )) ) 1 ) (14) At the present state of the literature, maintaining the proposed constants defined by (Murakami & Endo, 2023) (C* = 10 -4 ) and m* = 2), and assuming the fatigue limit ( ) for N=4x10 5 cycles, the predicted S-N curves are obtained and represented in Fig. 2.

Fig. 2. S-N models, Experimental curve vs Murakami equation.

To implement the optimisation algorithms, it was used the Software Python 3.10, more precisely the library SciPy 1.15.1; which allowed for finding the most suitable parameters C* and m* (equation 4). The chosen algorithms for this task were Nelder-Mead and SLSQP. The cost function to be minimised was the -R2, and the data used to find the optimal C and m was the experimental fatigue data from (Alves et al., 2024). The initial guess and the range of C and m were defined according to the previous work presented by (Alves et al., 2025), which was based on an improved solution that avoided the algorithms from diverging from an optimal solution. It is important to note that the initial guess between algorithms was different, while Nelder-Mead performed well for points near the optimal solution, for the SLSQP, a further apart point was needed for the convergence, and therefore, further analysis of a suitable initial guess had to be chosen. 5. Results and Discussion The use of different areas of defects, for the constant values of C* and m*, proposed by (Murakami & Endo, 2023), revealed to be conservative regarding the methodology used in this research. Furthermore, it was evident that a better description was needed when considering the final area of defect of 2000 μm. Although this is somewhat expected since by increasing the size of the maximum allowable crack size, the ratio (a f /a i ) increases, leading to a higher value of stress for the same number of cycles. Translating what is seen in real-world applications that for lower safety factors, higher stress is acceptable. By using both optimisation algorithms (Nelder-Mead and SLSQP) to find the optimal values C and m , it was possible to describe the fatigue behaviour of a Ti-6Al-4V manufactured by SLM, based on the defects gather through