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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stability (McClung et al., 1991; Antunes and Rodrigues, 2008). Neither of the two, however, can adequately describe the continuous fatigue process. The discontinuous nature of crack propagation, where propagation increment ∆ a is defined by element size l e , can be considered one of the limitations in the current framework of finite element crack propagation simulations (Antunes and Rodrigues, 2008). Node release at maximum load is chosen here. In order to reach a stable state before load reversal when crack propagation takes place at maximum load, an additional step of maximum load after node release is applied (Camas et al., 2018). Fig. 1 depicts the repeating loading sequence. The number of load cycles between the node releases determines the crack growth rate for a given element size. In finite element simulations, crack growth rate is orders of magnitude higher than with real cracks (Antunes and Rodrigues, 2008). It is currently infeasible to target realistic crack growth rates in simulations due to the computational cost. Usually at least two load cycles are recommended (propagation at every other load cycle) as propagation at every load cycle can yield artificially elevated closure levels due to the transient plastic strain field (Singh et al., 2008; Antunes and Rodrigues, 2008). Four load cycles are applied here between node releases to find a balance between su ffi cient plasticity stabilization and computational cost. The output quantities are the vertical displacement u y of all crack flank nodes behind the current crack tip, and the maximum nominal stress intensity factor K max . Crack profiles and opening levels are derived from the nodal displacements in the last opening step of each loading sequence (Fig. 1). K max is also evaluated at this step from a contour integral encasing the structured mesh section. Many assumptions are required to establish a computationally feasible crack propagation model. The major limi tations of the model include the following: microstructural heterogeneity and other closure mechanisms are not mod eled, defect is modeled as a perfect spherical cavity, crack propagation is incremental, propagation rate is orders of magnitude higher than in real cracks, crack tip is infinitely sharp, and crack propagates in a perfectly circular shape. By enquiring the displacement of each node behind the crack tip, a crack profile can be plotted, providing informa tion of not only the crack tip opening displacement (CTOD) values but the opening displacement of the entire crack flank. This can be done at any point during crack propagation to also obtain the crack profile development. Fig. 3 presents the crack profiles at maximum load for the analyses with constant amplitude loading, single overload and single underload, respectively. Here the crack profile is drawn on every other crack tip position to reduce clutter. As expected, crack opening displacement increases monotonically with crack length at all points on the crack flank. The over / underload can be seen to produce a discontinuity in the crack profile in all curves after crack length a / r = 0 . 5, where the loading irregularity was applied. The opening (or closing) level measures the strength of premature crack closure. It is commonly defined as the frac tion σ op /σ max (Antunes and Rodrigues, 2008), where σ op is the nominal stress level at which the crack opens. Defining σ op is not always straight-forward. In present work, the node contact criterion is used; when the y -displacement of a crack surface node is zero, the crack is closed in the vicinity of that node. The precise opening point is obtained with a linear regression from the linear part of the time–displacement response in an opening step. Said response is linear for approximately the first half of the opening step, after which e ff ects of plasticity accelerate the nodal displacement. It is reported in literature that no significant di ff erence is observed in opening levels between node release at minimum or maximum load (McClung et al., 1991; Solanki et al., 2003). However, slightly lower closure levels were consistently obtained in the present study when the crack was allowed to propagate at minimum load, versus 3.2. Opening level 2.6. Output data and model assumptions 3. Results and discussion In this section, overload (OL) and underload (UL) results are compared to a constant amplitude (CA) reference. 3.1. Crack profile