Discussion of fracture paper #30 - Weight functions, cracks and corners

Weight functions are practical tools in linear elastic systems where several discrete or continuously distributed sources cause something, deformation, stress, or related stuff. In linear fracture mechanics, as also in the object of this blog, weight functions are used to calculate stress intensity factors. If the load is divided into discrete or continuous separate or overlapping parts which each gives a known contribution to the stress intensity factor, i.e. has a known weight, calculation for new loads may be reduced to simple algebra instead of extensive numerical calculations. This is of course something that is frequently used by all of us. It is just the expected result of linearity. However, in the paper:

"Asymptotic behaviour of the Oore-Burns integral for cracks with a corner and correction formulae for embedded convex defects" by Paolo Livieri and Fausto Segala in Engineering Fracture Mechanics 252 (2021) 107663, https://doi.org/10.1016/j.engfracmech.2021.107663

an important step is taken. The authors show how weight functions can be used for 3D cracks with irregular shaped cracks including sharp corners. As the reader probably knows, there are exact solutions for simple straight cracks, penny shaped cracks and its inverse, a ligament connecting two half-spaces. The geometries of all these are 2D but with the application of arbitrary point forces acting on the crack surfaces the problem becomes 3D. There may be more such solutions unknown to me, but for virtually all realistic cases we are referred to numerical methods. Closed form solutions are indeed rare, but weight functions offer direct access to exact formulations that may bring about analytical simplifications, such as a variety of series expansions, direct integration of extracted singularities (cf. J.R. Rice 1989) and much more. It opens a world of clarity that never comes about when dealing with numerical models.

The school book part with known weight functions and arbitrary load is readily understood, but that will be blown away while real cracks or material flaws usually are neither plane nor perfect circles. The paper gives a nice introduction to the subject and provides a manual for how to deal with the problematic crack edge irregularities including sharp outward corners. It involves approximations which of coarse may lead to possible inaccuracies.

The lack of exact solutions is a two-edged sword. It is an enticement that motivates studies but with the consequence that there is nothing exact to verify the result against. The authors compare with their own and others' FEM results. I have no problem with that. Only the differences are of the order of what one expects from FEM which leaves us in limbo, not knowing if the weight function method is much better or twice as inaccurate. It is a consolation though, that the differences are a few percent only.

The authors suggest the method also requires comparison with experimental data. I agree with that in general, but I think it falls outside the scope of the study. It seems to me that it is a model selection problem, that has another context. I think it is good enough that the numerical model provides reliable results that are consistent with the mathematical description, i.e. the theory of elasticity.

The results are limited to convex corners, why this is so I could not understand. Is not a slight change of approach sufficient to include also concave corners?

It seems likely that the stress intensity factor becomes unlimited if an inward corner is approached from each side of the corner. In a real world, the stress intensity factor increases until something that blurs the picture pops up, e.g., that the corner is not sharp but rounded off or that the material is not linear elastic after all.

The situation slightly resembles a large crack with a tip very close to a free surface. For that, two different complete series expansions can be matched together with analytic continuation in a region where they both converge. The remote description is a power series with 1/r as a leading term. Around the crack tip a Williams series converges in circular region around the crack tip. Because of the different descriptions the stress intensity factor is given by the coefficient of the 1/r term times a-1/2, where the distance between the crack tip and the free surface, a, is the only length scale available. It is the direct result of dimensional considerations.

Perhaps an ansatz based on something that gives a singularity at the corner of the crack. The comparison with the crack approaching the free surface did not give any ideas per se. Possibly could it help if there was an unsharp corner. I feel that I am on thin ice here. Perhaps someone can give a hint of what to do. The first question is why did the authors exclude the concave corners.

Finally we hope that those who are interested in the subject would comment or contribute with personal reflections regarding the paper under consideration.  

Per Ståhle