PSI - Issue 76

Kimmo Kärkkäinen et al. / Procedia Structural Integrity 76 (2026) 11–18

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2. Modeling

This section describes the numerical crack growth model for evaluating plasticity-induced crack closure. The model configuration is identical to Ka¨rkka¨inen et al. (2024), and therefore only necessary details are briefly described here. Two di ff erent specimen geometries are used; an axisymmetric cylindrical specimen with an internal penny-shaped crack, and a plate-like specimen with a through-plate middle crack. Both specimens are investigated using the same 2D geometry and mesh by utilizing di ff erent sets of boundary conditions. For the plate-like specimen, both plane stress and plane strain conditions are considered, leading to a total of three di ff erent model configurations: axisymmetric (AX), plane stress (PS), and plane strain (PE). All specimen dimensions, diameter / width and length, are 2 mm. The sharp initial crack in the middle has a radius or half length of 40 µ m. A symmetry boundary condition is imposed on the bottom surface of the model and modified during the analysis to allow for crack propagation. A hard frictionless contact is defined between the crack surface and an analytical rigid surface corresponding to the other crack flank. A fine structured mesh consisting of fully integrated linear quadrilateral elements is used near the crack path, and the rest of the model is freely meshed with the same element type. A minimum element size of 0.5 µ m is chosen in accordance with the literature recommendations, being roughly 1 / 10 of the theoretical forward plastic zone size.

Fig. 2. Finite element model details. (a) Geometry. (b) Loading pattern. (c) Material model.

A continuum material model with linear kinematic hardening is defined corresponding to a high strength steel. Material parameters are Young’s modulus E = 210 GPa, Poisson’s ratio ν = 0 . 3, hardening ratio H / E = 0 . 05, yield strength σ y = 772 . 1 MPa, and Vickers hardness HV = 300kgf / mm 2 . An example of the material model behavior in a uniaxial stress state is presented in Fig. 2(c). Loading consists of zero–tension ( R = 0) constant amplitude cyclic stress applied uniformly to the top surface of the model. To produce results representative of near-threshold conditions, the stress amplitude is chosen to correspond to the fatigue limit determined by the Murakami–Endo model for internal defects (Murakami, 2019b),

2

( √ area ) 1 / 6

α , α = 0 . 226 + HV × 10 − 4 .

1 . 56( HV + 120)

1 − R

(1)

σ w =

For the cylindrical specimen, this results in a stress amplitude of σ = 269 . 69 MPa. For the plate-like specimen, the loading is simply scaled by a factor of 2 /π to produce a similar stress intensity, resulting in a stress amplitude of σ = 171 . 69 MPa. These values also serve as a reference for the fatigue limit results provided in Section 4. Crack propagation is realized by releasing the current crack tip node at maximum load every fourth load cycle (see Fig. 2(b)). One additional step of maximum load is given after node release. The total crack extension is chosen to be ∆ a = 30 µ m to allow for stabilization of plasticity-induced crack closure, corresponding to crack growth through 60 elements. The result, crack closure ratio, U , is obtained from the final loading step per crack advance.