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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3. Theoretical interpretation of the fatigue notch sensitivity Near a notch, stress concentrations are found during testing. Consequently, notched specimens break after a shorter fatigue life compared to unnotched specimens at the same level of external load. In the frame of conventional elasticity, stress concentrations are usually described by Peterson’s stress concentration factor. The stress concentration factor K t of an elastically deformed sample is defined as the peak stress experienced by a body in the vicinity of a notch divided by a reference stress 0 found at the smooth part of the body: (1) (2) where ℎ is the endurance stress limit of a smooth sample, while ℎ is the endurance stress limit of a notched sample whose stress was measured at its smooth region. Thus, the fatigue notch sensitivity factor q is defined as (3) In general, the fatigue notch sensitivity factor is in the range 0 ≤ ≤ 1 . In the present investigation, the determination of a fatigue notch sensitivity factor is challenging because a strain gauge cannot be attached to the miniaturized sample. Instead, the strain was measured at the sample holder. However, the strain at the reference point does not agree with the elongation of the sample holder at the hole. In consequence, the modified K t values relating the strain at the sample notch to the strain at the measurement reference point of the sample holder are larger than the usual K t values. Nevertheless, the results may be displayed in the style of Fig. 4 relating the evaluated stress at the sample notch to the lifetime. In conclusion, a fatigue notch sensitivity factor of q = 1 would correspond to a diagram where the Wöhler curves of differently notched samples are covering each other. However, sample types E and F show much higher stress levels compared to types A, B, C, and D, and therefore, sample types E and F have a fatigue notch sensitivity factor q clearly below 1. 3.1 Strain gradient elasticity In the following, it is demonstrated that the reduced fatigue notch sensitivity can be explained in the framework of strain gradient elasticity. This approach assigns additional material bending stiffness to small-scale structures. In conclusion, the Cauchy stress at the root of a sharp notch is reduced, and the replotted curves for sample types E and F show the same trend as the Wöhler curves of sample types A, B, C, and D. The conventional theory of elasticity does not include intrinsic material length scales, and therefore, it is incapable of explaining mechanical size effects. Within strain gradient elasticity, however, mechanical size effects in the sense that smaller is stronger can be described. The same theory also predicts a reduction of stress concentrations at a notch. The energy density of the strain gradient elasticity proposed by Mindlin (1964) reads as (4) where summation is carried out over repeated indices. Further, (5) is the second gradient of the displacements u k , while x i are the coordinates and ε ij is the linear strain tensor. Λ and Μ are standard Lame constants, while the parameters A 1 through A 5 are the material parameters of strain gradient theory On the other hand, the fatigue notch factor is defined as