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Discussion of fracture paper #9 - Crack tip modelling

ESIS's picture

Dear Reader, 

I recently took over as the ESIS blog editor. Being the second in this baton relay, I will do my best to live up to the good reader expectations that has been established by my precursor, who is also one of the instigators of the blog, Wolfgang Brock. 

I did not follow the blog in the past. That I regret now that I go through the previous blogs. Here I discover many sharp observations of new methods and concepts paired with a great ability to extract both the essential merits and to spot weaknesses. Much deserve additional studies to bring things to a common view. We are reminded that common views, often rightfully, but not always, are perishable items.

Paper 9 in this series of reviews concerns phenomena that occur when a crack penetrates an interface between two materials with dissimilar material properties. In the purely elastic case it is known that a variation of Young’s modulus along the intended path of a crack may improve the fracture resistance of inherently brittle materials. If the variation is discontinuous and the crack is about to enter a stiffer material the stress intensity factor becomes unlimited with the result that fracture will never happen. At least if the non-linear region at the crack tip is treated as a point. To resolve the problem the extent of the non-linear region has to be considered.

The selected paper is: Effect of a single soft interlayer on the crack driving force, M. Sistaninia and O. Kolednik, Engineering Fracture Mechanics Vol. 130, 2014, pp. 21–41

The authors show that spatial variations also of the yield stress alone can improve the fracture resistance. They find that the crack tip driving force of a crack that crosses a soft interlayer experiences a strong dip. The study is justified and the motivation is that the crack should be trapped in the interlayer. The concept of configurational forces (a paper on configurational forces was the subject of ESIS review no. 7) is employed to derive design rules for an optimal interlayer configuration. For a given matrix material and load, the thickness and the yield stress of a softer interlayer are determined so that the crack tip driving force is minimised. Such an optimum configuration can be used for a sophisticated design of fracture resistant components. 

The authors discuss the most important limitations of the analysis of which one is that a series of stationary cracks are considered instead of a growing crack. The discussion of growing versus stationary cracks is supported by an earlier publication from the group. Further the analysis is limited to elastic-ideally plastic materials. A warning is promulgated by them for directly using the results for hardening materials.

The paper is a well written and a technically detailed study that makes the reading a good investment.

The object of my discussion is the role of the fracture process region in analogy with the discussion above of the elastic case. The process region is the region where the stresses decay with increasing straining. When the process region is sufficiently small it may be treated as a point but this may not be the case when a crack penetrates an interface. The process region cannot be small compared to the distance to the interface during the entire process. In the elastic case the simplification leads to a paradoxical result. The main difference as compared with the elastic case is that the ideally plastic fields surrounding a crack tip at some short distance from the interface have the same characteristics as the crack that has the tip at the interface, i.e. in the vicinity of the crack tip the stress is constant and the strain is inversely proportional to the distance to the crack tip. This means that the distance between the crack tip and the interface do not play the same role as in the elastic case. A couple of questions arise that perhaps could be objects of future studies. One is: What happens when the extent of the process region is larger than or of the order of the distance to the interface? If the crack is growing, obviously that has to happen and at some point the fracture processes will probably be active simultaneously in both materials. The way to extend the model could be to introduce a cohesive zone of Barenblatt type, that covers the fracture process region. The surrounding continuum may still be an elastic plastic material as in the present paper.

A problem with growing cracks is that the weaker crack tip fields does not provide any energy release rate at a point shaped crack tip. Would that limitation also be removed if the finite extent of the process region is considered?

With these open questions I hope to trigger those who are interested in the subject to comment or contribute with personal reflections regarding the paper under consideration.


Per Ståhle
Professor of Solid Mechanics
Lund University, Lund


Dear Per

I think, you raised very interesting questions in your comments to our paper. 

Let us first consider a linear elastic material with inhomogeneity of the Young's modulus E, i.e. E exhibits a jump at the interface. Theory predicts that, for a crack ending directly at the interface, the crack driving force becomes infinite, if E decreases in the crack growth direction. The crack driving force becomes zero if E increases in the crack growth direction. 
As you write, this is unrealistic, and one should introduce a non-linear region around the tip. How large should such a region be? A related question is, how sharp can a real bimaterial interface be?
A region not less than an atomic distance is maybe reasonable.

I am not sure whether I understand correctly your question in the case of yield stress inhomogeneity. If we take the length of the process zone l_proc as being proportional to the crack tip opening displacement (with the proportionality constant of the order of 2), then we have considered in the paper also cases where the distance L between crack tip and interface is smaller than l_proc. I do not see a big problem here. Do we overlook something?

Another problem is - and maybe that is also what you had in mind - that not only the crack driving force, measured in terms of J_tip, changes when the crack tip approaches the interface. Also, the crack growth resistance R will, in general, change. You know, the crack extends, if J_tip >= R. How can we model the change in R? Here we have had, as you also proposed, the idea of applying the cohesive zone model since this model allows us to with an intrinsic fracture resistance of the materials, prescribed by the cohesive energy. The materials left and right of the interface can then have different material properties and different characteristic cohesive zone parameters (c energy and cohesive stress). 
We have started with the analyses, the first results are already available, but not yet published.

Best regards,

Otmar Kolednik

(This comment was submitted by Professor Kolednik on November 10, 2014. Accidently it was removed for a while. I am sorry about that. PS)

ESIS's picture

Dear Otmar,

Thank you for the comment. You are right, there is also the interface. In the case of a penetrating crack it might have a width that has to be considered. So two simplifying circumstances arise.  Either the fracture process zone is small compared to the width of the interface. Then I guess the crack tip can be treated as a point with a prescribed KIc-tip, critical J_tip or whatever that can quantify the critical state of the crack tip. Opposed to this, the width of the interface may be much less than the size of the fracture process zone. Then the details of that process zone needs to be resolved. Unfortunately there are also general cases when neither the crack tip or the interface can be treated as perfectly sharp.

About the yield stress inhomogeneity, I am a bit uncertain myself what I meant. Of course as you write there may be a jump in yield stress. Then if the fracture process is ductile with void nucleation, growth and coalescence, a sharp crack tip model attempt to summarise all stages with a single parameter. The tip will be in either material A or material B. On the other hand, if the real crack is penetrating the bi-material interface, first there has to be void growth and coalescence in material A and void nucleation in material B. Later, the void growth and coalescence occur in material B and finally the entire process zone is in material B. This intermediate state will perhaps give a variation of the fracture toughness that otherwise is identical in both materials.

One might imagine two ductile materials with same elastic-plastic properties and same fracture toughness. Say that the only difference between the materials is that one has many and fragile inclusions while the other material has few and strong inclusions. Then the first material has its toughness from the extensive energy that is required for void coalescence, and the second material requires much energy for void nucleation. As you suggest, a cohesive zone model might be the solution. A sharp crack tip model would not reveal any difference between the materials while a proper cohesive zone model would. The material model for the continuum that surrounds the cohesive zone can still be elastic-plastic. I guess that would not be too complicated since you already have a model developed for that part. That would have the reward that you would be able to distinguish between energy dissipating in the stable elastic-plastic continuum and the unstable material fracture process zone modelled by the cohesive zone.

I am looking forward to your coming publication and the next episode of this story.


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