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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closure phenomena, such as crack tip blunting and partial closure (Paris et al., 1999; Borrego et al., 2003). It is well established experimentally that crack propagation is hindered following an overload and accelerated after an underload. More precisely, crack growth first accelerates after an overload, but then rapidly decelerates, and slowly returns to a normal rate after some propagation (Mlikota et al., 2017). In experiments, a beachmark corresponding to the crack front location at the time of the applied loading irregularity can be observed (Vosikovsky and Rivard, 1981). The extent of the underload e ff ect is said to be lesser than that of the overload (Liang et al., 2022b). Many accounts of numerical investigation of over / underload e ff ects also exist (Ellyin and Wu, 1999; Pommier and Bompard, 2000; Baptista et al., 2017), usually considering plasticity-induced crack closure. Alongside being able to explain, for example, e ff ects of stress ratio, plasticity-induced crack closure is an intuitive explanation for the e ff ect of over / underloads on the fatigue crack propagation (Aguilar Espinosa et al., 2013). After an overload, the compressive residual stress field in front of the crack tip is more intense, which translates to a more significant plastic wake behind the crack tip once it propagates into the overload plastic zone. A similar but reversed mechanism can be imagined for underloads, where the tensile plastic strain in the crack wake is reduced by reversed plasticity. Existing papers on the phenomenon have mostly considered long cracks where e ff ects of crack closure have had a su ffi cient crack propagation length to stabilize. Some experimental analyses also consider overload interactions of short cracks (Changqing et al., 1996; Song et al., 2001). These works conclude that the retardation e ff ect following an overload is similar, if not strengthened with short cracks. Suresh and Ritchie (1984) however have stated the opposite, that short cracks would not experience the retardation / acceleration e ff ects to the same degree as long cracks. It seems unclear how these mechanisms operate in short cracks, and to the authors’ knowledge there are currently no papers considering the numerical modeling of short, interior defect-initiated crack propagation under a single over / underload. The current analysis considers an interior defect-initiated crack, making use of rotational symmetry; plane strain conditions prevail in this situation. It is known that plasticity-induced crack closure is much stronger in plane stress than in plane strain (McClung et al., 1991), which increases the uncertainty of how single loading irregularities will a ff ect the levels of plasticity-induced closure. In order to shed light on the relationship between plasticity-induced crack closure and empirically observed crack growth rate transients in short, interior defect-initiated crack propagation under overloads and underloads, an ABAQUS finite-element model, described in the next section, is employed.

2. Modeling methodology

The chosen finite element modeling strategy, largely similar to what is used for the simulation of surface defect initiated cracks in Ka¨rkka¨inen et al. (2023), is described in this section.

2.1. Specimen geometry

The finite element model represents a cylindrical test specimen with height and diameter equal to 2 mm. Rotational symmetry and half symmetry in the axial direction is used to reduce model size and computational e ff ort. An internal defect, a typical origin of failure especially in the very high cycle regime, is placed at the specimen’s rotational symmetry axis and modeled as a spherical cavity with a radius of r = 40 µ m. The resulting model geometry is a 2D square with a quarter sector of a circle cut out from the corner at the intersection of the symmetry axes. The same specimen geometry, apart from the defect location, was used in Ka¨rkka¨inen et al. (2023). Here the internal placement of the crack-initiating defect allows for faster simulation with a rotationally symmetric model, whereas a 3D model was required with the surface defect. A depiction of model geometry and finite element mesh is presented in Fig. 1. Contact between the other half of the crack is simulated by placing an analytically rigid surface in the symmetry plane. Contact formulation between the crack surface and the rigid surface consists of hard contact relationship and node-to-surface discretization method.

2.2. Finite element mesh

The 2D geometry is discretized to a fine, structured mesh of linear quadrilateral elements (CAX4) near the crack path. The rest of the geometry is meshed freely with linear triangular elements (CAX3). The finite element mesh used in the simulations is presented in Fig. 1. The structured mesh section is sized to encase the plastic zone of the