PSI - Issue 57

Kimmo Kärkkäinen et al. / Procedia Structural Integrity 57 (2024) 271–279 K. Ka¨rkka¨inen et al. / Structural Integrity Procedia 00 (2023) 000–000

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4

Fig. 2: Constitutive cyclic behavior of the defined material model with kinematic hardening in a uniaxial stress state.

shows particularly good correlation with Vickers hardness according to Pavlina and Vantyne (2008), and the use of this relation was chosen for the sake of simplicity. Bilinear material behavior, i.e., linear hardening is defined for the material model. The hardening ratio is H / E = 0 . 05, where Young’s modulus is E = 210 GPa and hardening modulus H = 10 . 5 GPa. Hardening behavior is per fectly kinematic. As strains remain relatively low in the present simulation, ultimate tensile strength is not assigned; hardening behavior remains linear for the entire plastic strain range. An example of the material model behavior in a uniaxial stress state is presented in Fig. 2. As plasticity-induced crack closure has the greatest significance near fatigue limit, loading is chosen to correspond to the fatigue limit of the Murakami–Endo model (Murakami, 2019). This empirical–analytical fatigue model is shown to predict defect-initiated fatigue limits remarkably well for both internal and surface defects (Scho¨nbauer et al., 2021; Murakami, 2019). For smooth steel specimens in high and very high cycle fatigue, initiation from small defects, such as non-metallic inclusions, is observed to be the primary fatigue crack initiation mechanism (Murakami and Endo, 1994; Murakami and Beretta, 1999; Scho¨nbauer et al., 2021). As the present work considers short fatigue cracks initiated from a microscopic internal defect, a situation very well within the application range of the Murakami–Endo model, this model gives a suitable estimate for the fatigue limit. The applied stress amplitude is the fatigue limit when controlled by crack propagation from an internal defect, given by Eq. 3. The stress ratio considered in present work is R = 0. This results in a stress amplitude of 269.69 MPa, maximum and minimum loads being 539.39 MPa and 0 MPa, respectively. 2.4. Loading

2

( √ area ) 1 / 6

α , α = 0 . 226 + HV × 10 − 4

1 . 56( HV + 120)

1 − R

(3)

σ w =

In present work, the e ff ect of loading irregularities is investigated by applying a single overload or underload at the halfway of crack propagation, i.e., at a / r = 0 . 5. The magnitude of the overload is chosen to be 1.3 times the base maximum load (701.20 MPa), avoiding global yielding. The underload is equal in magnitude but opposite in direction (-701.20 MPa), which results in a significantly larger underload ratio compared to the overload ratio. As a crack tip stress singularity is present in tension but not in compression, the crack response to an underload is much less severe. Current over / underload levels are chosen to produce a pronounced e ff ect of the loading irregularity in both cases.

2.5. Crack propagation scheme

In present analyses, the crack propagates to a final crack length a = r . Propagation is achieved by consecutively releasing the symmetry boundary condition of the current crack tip node, which is the most commonly used method in literature (Oplt et al., 2019; Camas et al., 2018; Sta¨cker and Sander, 2017). Incremental propagation is usually set to take place at maximum load, which is physically more intuitive, or minimum load, which can improve numerical