PSI - Issue 47

Gianmarco Villani et al. / Procedia Structural Integrity 47 (2023) 873–881 G. Villani et al. / Structural Integrity Procedia 00 (2023) 000–000

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Gilchrist and Smith (1991) predicted the fatigue crack growth by considering various points along the front, with growth occurring normally from the crack front. In other investigations Zhou (2006); Song and Shieh (2004); Paul (1995); Fang et al. (2006); Choi and Choi (2005), the strain energy density factor (SEDF) approach, also known as Sih’s law Sih (1979), was employed to analyse the fatigue crack growth. The use of the SEDF approach in these studies enabled the consideration of the influence of mixed mode and stress ratio R. Anyway, the SEDF approach is based on the stress intensity factors computation because the strain energy density factor is derived by the three stress intensity factors for mode I, II and III. Hachi et. all developed a hybrid weight function method in order to calculate the stress intensity factors applying the method to an infinite cracked body and comparing the obtained results to the SEDF approach Hachi et al. (2010). In this work, we propose another method with the aim to predict the crack shape during the propagation, based on an energetic approach. The method is based on two main hypotheses: one regards the crack shape during the crack growth and the second one regards the driving force that governs the crack during the propagation. We assume that the material is elastic and that the crack is initially elliptical and maintains an elliptical shape during the crack evolution. The minimization of the elastic strain energy has been considered as driving force for the crack evolution. The proposed method has been applied to three di ff erent cases: a penny shape cracked infinite body, an infinite body with an elliptical crack and a surface cracked plate loaded in bending. A Matlab ® routine has been carried out to automatically performs the numerical computation, linking a Matlab ® script to manage the elastic strain energy evaluation and the crack shapes, and an ANSYS Mechanical APDL ® script to perform the simulations iteratively. The obtained results have been compared to literature ones obtained by Refs. Lin and Smith (1999b); Hachi et al. (2010), allowing to conclude that the calculated crack propagation paths describe with good accuracy the evolution of elliptical cracks.

2. Specimens details and mesh

The crack growth numerical analyses to determine the crack shape were conducted on three di ff erent typologies of specimen:

• A penny shape cracked infinite body • An infinite body with an elliptical crack • A finite body with a surface crack subjected to bending moment

Fig. 1a shows the penny shape cracked infinite body and the modelling strategy. For the FE model, three symmetries were considered. Fig. 1b shows the faces where symmetry constraints were applied. The simulation of the penny shape cracked infinite body was carried out considering an imposed displacement; thus, the bottom surface was subjected to a displacement δ . A similar modelling approach was performed to study the behaviour of an initial elliptical crack (Fig. 2). The surface cracked plate was modelled considering a circular initial crack. Fig. 3a shows the specimen and M represents the applied bending moment. Fig. 3b shows the faces where the symmetry constraints were applied.

3. Methodologies and numerical procedure

The energetic approach to calculate the crack propagation is based on two main hypotheses:

• The initial crack is elliptical and during the crack propagation the shape of the crack remains elliptical (but the semi-axes ratio may vary during the propagation) • The method considers the elastic strain energy calculated for equilibrium states. The crack propagation is driven by the minimization of the elastic strain energy The first hypothesis implies that during the crack propagation, the semi-axes change (even in di ff erent amount, modifying the aspect ratio), maintaining the elliptical geometry. Indeed, starting from the initial elliptical crack shape characterised by the semi-axes a 0 and b 0 and imposing an area increment ∆ A , it is possible to consider several virtual