PSI - Issue 74

Mitra Delshadmanesh et al. / Procedia Structural Integrity 74 (2025) 9–16 Mitra Delshadmanesh / Structural Integrity Procedia 00 (2025) 000–000

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tensor were used as generalized DOF, leading to elements with 36 DOF. The strain at the barycenter of the tetrahedra was determined with first-order interpolation functions. Further, the strain gradients were evaluated from differences in strains at the barycenter of neighboring elements. For this purpose, a method of sequential interpolation was introduced, which utilizes secondary grids in four tetrahedral sub-elements arranged by analogy to a three-dimensional Clough-Tocher scheme. Every sub-element is attached to the barycenter and to 3 corner nodes of the main tetrahedron. The interior of the sub-elements is interpolated with second-order polynomials of displacements to determine the strain gradients. Notice that user elements themselves are not shown in ABAQUS graphics. Hence, postprocessing of user elements was achieved with the help of conventional dummy elements of very small stiffness, which were connected to the same nodes. Thus, the strain gradient material parameters were subjected to constrained optimization by considering condition (6), leading to values of A 3 = - 140 N and A 4 = 280 N. The additional bending stiffness of this model leads to a reduction of the von Mises stress at the notch. The stress distribution around the notch with a 125 µm radius of a sample with a lifetime of 4.85•10 6 loading cycles to failure, simulated according to strain gradient elasticity, is shown in Fig. 6.

Fig. 6. Distribution of von Mises stress [MPa] in a sample with 125 µm notch radius simulated according to strain gradient elasticity. The vibration load assumed in this harmonic analysis is related to a sample life of 4.85•10 6 loading cycles to failure. All sample types were simulated with strain gradient theory to replot Fig. 4. The results are shown in Fig. 7. 4. Discussion The present study shows that even though notched specimens have shorter fatigue lives than unnotched samples, the reduction in lifetime of samples with a sharp notch is less than expected according to the predictions based on the Peterson stress concentration factor. This effect could well be explained by strain gradient elasticity. In conclusion, the endurance limit of the stress at the notch was for all sample types smaller than the material's yield strength of 262 MPa, which was measured by Zare Ghomsheh et al. (2020). For completeness, it should be mentioned that the fatigue lives observed in ultrasonic testing are, in general, longer than observed at smaller deformation rates, as pointed out by Geilen et al. (2020). But this argument equally applies to notched and unnotched specimens. Furthermore, it should be said that the fatigue lives of miniaturized samples are mainly governed by the time to crack initiation, because the period of crack propagation is short. Nevertheless, the results of the Finite Element simulation performed according to strain gradient theory are plausible.