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Discussion of fracture paper #12 - Crack paths and fracture process region autonomy
Cracks typically follow paths that locally give a mode I crack tip load. At mixed modes crack are extended via a kink in a direction that locally restores mode I. In isotropic materials this is known to more or less, true for static and dynamic loads. Exceptions are cracks that are subjected to high compressive load, e.g., at contact between train wheels and rails or at cracks caused by seismic movements. Other exceptions are cracks growing in anisotropic materials, at grain boundaries or other weak, or by deformation weakened, interfaces.
The recently published
is an interesting paper about the calculation of mixed mode loads and crack paths by use of a combination of several concepts for crack tip modelling. The developed method has general applicability in that it includes cracks that pass, join or deflect from interfaces, cracks at bifurcation points where three materials meet, and of course the crack paths embedded in homogeneous materials. A scheme is presented that uses two slightly altered local meshes to obtain the variation of the energy release rate due to a small variation of the crack path. The M-integral by Yau, Wang and Corten, J. Appl. Mech., 1980 for separation of mode I and mode II is used. The paper is nicely completed with a demonstration of a crack propagation framework, which combines the developed methods. The result is a convincing simulation of crack growth through a composite material. The path that maximises the strain energy release rate relative to the toughness, is followed.
When the fracture processes are confined to a small region it may be safe to use a sharp crack tip. However, occasionally it leads to an unreasonable behaviour, such as when the energy release rate disappears as the crack tip passes through a bimaterial interface from a weaker to a stiffer material (cf. discussion of paper 9 in this series). Similarly, let's say that a crack meets a conceivable branching point with two branches or paths to chose between and both paths are having equal loading and equal toughness. This seems to be a dead heat. However, say that initiation of the fracture processes need sufficient hydrostatic stress and sufficient subsequent deformation to complete the fracture and that the relation between these quantities are different along the two paths. Then even though the toughnesses are equal, the growing crack is likely to follow the path that first allow initiation of the fracture processes and the other path will never be activated. Perhaps there are exceptions but in general it seems to me that a crack tip model with more details is needed for these cases.
Cases when cracks deflect from a weak interface are, I believe, similarly problematic. Whether a crack will follow a weak plane under a mixed mode load or kink out of that plane should to a large extent depend on the affinity to initiate a fracture process outside the interface.
I understand that the paper is concerned with indivisible fracture toughness which is excellent, but in view of the sketched scenarios above, I wonder if the model can be extended to include modelling of the process region with a finite physical extent, e.g. by using a cohesive zone model, that provides a two parameter model for the process region. One difficulty that I immediately come to think of is that the strain energy singularity is annulated by the cohesive stresses so that the M-integral possibly will fail. Still, if the foremost part of the process region, i.e. the tip of the cohesive zone rather than the crack tip, is the path finder then maybe a stress criterion could be a suitable candidate. Are there other possibilities? Could the point shaped crack tip be kept while using a stress criterium at some fixed distance ahead of the crack tip? Or would "fixed distance" per se require process region autonomy?
Per Ståhle
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Thank you for the thoughtful
Thank you for the thoughtful review of our recent paper. You bring up many different situations that might be encountered when a crack interacts with an interface. The methods developed in the paper seek to lay a foundation upon which we can incorporate more complicated scenarios, such as those you described above. For readers who have not viewed the paper, I will first provide some additional background information.
An important component of the framework involves a library of pre-created mesh patterns (referred to as templates) which are automatically inserted around the crack tip. The templates contain crack extension increments at many different angles, and they are incorporated into an existing FE model at a crack tip. The existing model can have up two 8 different material regions joining at the tip. For growth at each different angle of interest, there are two nearly identical templates, where the main difference is that in one of the templates the nodes along the extension increment are allowed to separate, while in the other template, the extension increment is simply represented in the mesh. These templates are automatically inserted and enable efficient acquisition of “before” and “after” FE analysis results. For the scenarios investigated in the current paper the GI and GII were obtained by considering the forces ahead of the crack tip in the “before” model and the corresponding opening displacements in the “after” model.
The situations you describe could be largely investigated by expanding the framework in two main areas: 1) extracting the criterion-specific stress/strain/displacement values from the FE results and 2) developing additional mesh template sets. First, thinking in terms of a fracture initiation criterion that requires a specific stress state or deformation, these values could be exported from the analysis results and automatically compared to the critical values. This would be a straightforward alteration to the existing code. To include a cohesive zone model, mesh template sets could be developed such that they included interface elements along the crack extension increment. The current implementation can use supplied interface toughness values that are dependent on the two adjacent materials. Different traction-separation relationships could likewise be applied depending on the combination of adjacent materials. Using the mesh templates to insert interface elements immediately ahead of the existing tip at any angle of interest, allows the crack to grow in any direction without having to follow a pre-determined path that contains interface elements. Additional work would be required to adjust the framework and existing FE code to properly handle the interface elements.
Overall, this methodology was developed to be expandable, so that it could be adapted to work with different fracture criteria and material properties. For any given multiple-material structure, appropriate criteria should be selected and incorporated into the framework when necessary.