PSI - Issue 79

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

329

= 1 + ( − 1 )

(9) Other interesting algorithm, with a different working principle is SLSQP, which uses the gradient to identify where to move through loss function, locally adjusting a quadratic equation, given by equation 10 (Fu et al., 2019). H k represents the Hessian Matrix. ( ) ≈ ( ) + ( ) ( − )+ 1 2 ( − ) ( − ) (10) By adjusting the quadratic equation, through gradient calculation it is possible to determine the direction in which it is possible to reduce the cost function, d, as represented by the equation 11 (Fu et al., 2019). = − −1 ( ) (11) To conclude the first iteration, the value that minuses the function must be updated, which happens through equation 12. Where is the step length. +1 = + (12) Therefore, the objective of this work is to propose a new equation by modifying the constants used by (Murakami & Endo, 2023), in equation 4, by recurring to optimisation algorithms, such as Nelder-Mead and SLSQP, fitted to the experimental data, to determine an optimal C and m that properly describes the fatigue data of a Ti-6Al-4V alloy manufactured by SLM.

Nomenclature area

Area of the defect

a

Crack size

a f a i C

Maximum Admissible defect’s size

Initial defect size

Constant of the material calculated by least squares method

C*

Universal Constant proposed by Murakami

c d h

Centroid

Search Direction

Distance between internal defect and sample surface

H k Hv

Hessian Matrix Vickers Hardness

l 0 m

Defect size transitioning from small crack to a long crack Constant of the material calculated by least squares method

m*

Universal Constant proposed by Murakami

n

Empirical Constant Number of cycles

N R

Stress ratio

R 2

Coefficient of determination

ΔK th

Threshold stress intensity factor range

x c ,x e , x i , x r

Simplex points Applied stress Fatigue Threshold

Σ

Δσ th Δσ W α , , , α

Fatigue limit

Empirical factor dependent on the variable hardness

Step Length