The behaviour of small fatigue cracks, and some basic modelling approaches for the phenomenon.

Abstract

The behaviour of small cracks has been explained by various different mechanisms (e.g. microstructural barriers, closure, stress state etc). The majority of models already developed are based on a single mechanism (e.g. crack closure, effect of microstructure), while the rest of the processes occurring simultaneously in the material are either considered to play a minor role or are included in fitting parameters obtained from experiments. Consequently the models are suitable only for materials and conditions similar to those used in the ‘fitting’ tests and thus hard to generalise and adapt for common engineering use. This work therefore aims to provide the foundation to develop a combination of the generalised crack closure model and the microstructural models including the effects of the varying microstructure and environment, respectively, to create a comprehensive and realistic model reflecting the major processes influencing small crack behaviour. A thorough review of the current literature and approaches applied for the modeling of small and short cracks is presented. The Distributed Dislocation Technique has been identified as a particularly suitable candidate for the modeling of the problem. Initial trials with the model proposed by Navarro and de los Rios however showed certain shortcomings in their model caused mainly by unjustified simplifications. The Distributed Dislocation Technique has been evaluated against an FE approach for the case of a kinked crack and shown to be potentially a very powerful tool for the solution of crack problems. Two numerical methods for solution of singular integral equations have been tested to allow for maximum accurate evaluation of plastic zone size (important in later stages of the model’s development). While the Lobatto-Chebyshev method offers a comfortable solution of the singular integral equations, its fixed calculation and collocation points invalidate it for the evaluation of plastic zone size. The Gerasoulis’ method is, on the other hand, more cumbersome to apply, thanks to the weight functions, which can be rather lengthy, but it’s flexible calculation and collocation points should enable more accurate evaluation of plastic zone size. Nonetheless, both methods demonstrated very good results for all problems, on which they were tested, even for a relatively limited number of calculation points. Incorporating microstructure more accurately into the model was approached by assuming (1) that a crack will grow into the adjacent grain on the slip plane with the highest resolved shear stress, if the resolved normal stress on the plane is tensile (2) that the slip would initiate from a source of dislocations in the grain. Five typical grain orientations (e.g. Cube, Goss, Brass, Cu, and S) were investigated and a number of possible scenarios were followed in the modelling. A dependence of the slip direction on the position of the source of dislocations and grain orientation was demonstrated. Although only a 2D consideration of the problem is needed for the model, the calculation of the resolved stresses was conducted in 3D and only the final slip direction was transformed to a 2D intersection of the slip plane with the surface of the specimen. Two ways to capture the presence of sources of dislocations in the structure were proposed and their suitability for the model should be further investigated in future work. The stochastic nature of the position of the dislocation sources regarding the crack tip can be represented either by a single dislocation source randomly placed, or by a number of dislocation sources randomly distributed in the grain. In the first case the calculation will be quicker and easier, as only one point will be considered, in the second case the reality of the problem will be represented better, but the calculation will be more complicated. The consideration of the microstructure in the model introduced an additional parameter into the model. Such microstructural qualities as width of grain boundaries, density of sources of dislocations, inclination of the material toward planar or wavy slip, etc. can be appreciated in the model by means of the position, distribution and also number of dislocation sources considered in the model. This offers a platform for very realistic representation of the small crack propagation. Nevertheless, the possibility of incorporation of these parameters into the model and their mutual interaction must be fully understood and thus, some simplifying assumptions will have to be introduced into any future developments of the model.

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ORCID for Philippa Reed: orcid.org/0000-0002-2258-0347

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