PSI - Issue 45

James Vidler et al. / Procedia Structural Integrity 45 (2023) 82–87 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

83

2

1. Introduction The concept of a breathing crack and the clapping mechanism, when a crack periodically opens and closes subject to a cyclic or harmonic excitation, are widely utilised in studies concerning contact acoustic nonlinearity (CAN) and nonlinear phenomena. The investigation of these phenomena is particularly important for the development of reliable non-destructive fatigue crack detection techniques (Broda et al. 2014). The simplest and most popular bilinear stiffness model of breathing crack considers an abrupt change in the stiffness of the structure when it is subjected to tensile and compressive loading. In tension, when the crack is open, the global stiffness of the structure is reduced; when the crack is closed, the stiffness is unaffected. This model has been widely applied in research into breathing cracks, for example, to study nonlinear vibrations of beams using both analytical and numerical approaches (Broda et al. 2014, Xu et al. 2021). In wave propagation problems the behaviour of a crack is often described as a mechanical diode: when the crack is closed during the compressive portion of the loading, it can be fully penetrated by the incident elastic wave; whereas when the crack is open, penetration does not occur over the crack surfaces (Broda et al. 2014, Xu et al. 2021). However, the clapping mechanism as well as the concept of breathing crack are not substantiated by the actual fatigue crack behaviour, as experimental observations indicate that in the absence of the applied load, fatigue cracks are partially closed (Elber 1970, Codrington & Kotousov 2007). To demonstrate the realistic features of fatigue crack behaviour under loading, consider a simple example of a through-the-thickness fatigue crack propagating in a plate. The stress state near the crack is very complex, and its examination requires 3D nonlinear Finite Element modelling coupled with crack advance procedures (Maia et al. 2016). However, the fatigue crack behaviour can also be analysed using the simplified strip-yield model presented by Dugdale (1960). This model is commonly used in fatigue research to account for the effects of plasticity. The propagation of the crack generates a wake of plasticity behind the crack tip, which leads to the plasticity-induced fatigue crack closure phenomenon illustrated in Fig. 1. It was first discovered and described by Elber (1970). Since this work the phenomenon has been investigated extensively. The first analytical model was developed by Budiansky and Hutchinson (1978) for a semi-infinite crack propagating under constant amplitude cyclic loading. The concept was generalised to different geometries and variable amplitude cyclic loading, and resulted in the development of several fatigue life-prediction codes, such as FASTRAN and NASGRO (Newman 1992). For the problem under consideration, the regions of the crack near the tips < | | ≤ are in contact (or closed); while the central part of the crack, | | ≤ , is opened as illustrated in Fig.1b, where is the half-length of the crack. Material near the faces of fatigue crack is stretched and forms the plastic wake, which can be of various shapes depending on the loading conditions. Ahead of the crack tip there are two regions: direct plasticity and reverse plasticity, which are shaped by the cyclic loading sequence (fatigue loading history) (Rose & Wang 2001, Codrington & Kotousov 2007, Maia et al. 2016). The aim of the current contribution is to demonstrate the characteristic features of the behaviour of fatigue crack under excitation loading using this simple model, which accounts for the actual mechanisms associated with fatigue crack propagation. It is believed that incorporation of more realistic fatigue crack behaviour can help to improve the methods for its detection utilising nonlinear phenomena associated with the CAN. 2. Problem formulation Several assumptions and simplifications are adopted. To simplify the problem all dynamic effects are disregarded. For fatigue cracks propagating in the high-cycle fatigue regime the size of the plastic zone ahead of the crack tip is typically much smaller than the crack length or, m and r ≪ , see Fig.1. Therefore, the influence of the shape of this zone on the central part of the crack, which is open, can be neglected. Moreover, for positive R-ratios (R = σ min / σ max , where σ min and σ max are minimum and maximum stresses of the cyclic loading) the plastic stretch remains approximately constant during loading and unloading, so it may be assumed that m ≈ r = . For example, Budiansky and Hutchinson model [5] for a semi-infinite crack shows that r / m is between 0.86 and 1, for R = 0 and 1, respectively.