PSI - Issue 45

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

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Similar results have been obtained for R > 0 and a self-similar growth of small cracks [6]. In accordance with the strip yield model (Dugdale 1960) the maximum stretch is = 8 ln sec π max 2 Y (1) where is Young ’ s modulus and Y is the yield stress. Eq. (1) indicates that if the fatigue crack grows under constant amplitude cyclic loading, the shape of the plastic wake is linear [6]. Therefore, the geometry of the problem can be simplified as two sharp wedges inserted into an infinite linear elastic plate, as illustrated in Fig.2a. The mathematical problem can be formulated in terms of edge dislocations: two edge dislocations with intensity located at = ± and the linear distribution of edge dislocation with intensity sign( ) × / represent the wake of plasticity (Codrington & Kotousov 2007). The unknown distribution of dislocations, y ( ), over the open part of the crack | | ≤ represents the open region of the crack. The latter distribution is such that the stress over this open region, = a , where a is the applied stress. The applied stress is understood to be generated by transducers, which are commonly utilised for non-destructive inspections. This stress is usually much lower than max in order to avoid inducing further damage in the structure through the inspection process. 3. Distributed dislocation solution The singular equation for the unknown dislocation density, y ( ), can be written as − ( , 0) ( 2+ 1) = − ∫ sign( ) − − +∫ ( ) − − − + + − (2) where is shear modulus and is Kolosov ’ s constant, which is defined as 3 − 4 for plane strain and (3 – ) / (1 + ) for plane stress conditions. Fig. 1. Propagating fatigue crack: (a) under loading and (b) unloading. All drawings are not to scale, normally and ≪ .