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# Discussion of fracture paper #14 - How to understand the J-integral when multiple cracks are growing at different rates

A nice demonstration of toughening by introducing multiple secondary cracking of planes parallel with the primary crack is found in the paper:

The 14th paper belong to the category innovative ideas leading to improved composites. We already know of combinations of hard/soft, stiff/weak or brittle/ductile materials that are used to obtain some desired properties. The results are not at all limited to what is set by the pure materials themselves. It has been shown that cracks intersecting soft material layers are exposed to elevated fracture resistances (see eg. the paper 9 blog). Differences in stiffness can be used to improve fatigue and fracture mechanical properties as found in studies by Surresh, Sou, Cominou, He, Hutchinson, and others. Weak interfaces can be used to diverge or split a crack on an intersecting path. A retardation is caused by the additional energy consumed for the extended crack surface area or caused by smaller crack tip driving forces of diverging crack branches.

A primary crack is confined to grow in a weak layer. The crack tip that is modelled with a cohesive zone remains stationary until the full load carrying capacity of the cohesive forces is reached. Meanwhile the increasing stress across an even weaker adjacent layer also develops a cohesive zone that takes its share of the energy released from the surrounding elastic material. At some point the cohesive capacity is exhausted also here and a secondary crack is initiated. Both cracks are confined to different crack planes and will never coalesce. The continuation may follow different scenarios depending on the distance between the two planes, the relative cohesive properties like cohesive stress, critical crack tip opening, the behaviour at closure etc. of the second layer. All these aspects are studied and discussed in the paper.

The investigators have successfully found a model for how to design the cohesive properties to obtain structures with optimal fracture resistance. Parameters that are manageable in a production process are the ratio of the cohesive properties of the different crack planes and the distance between the them. A theoretical model is formulated. With it they are able to predict whether or not the toughness of a layered structure can be increased by introducing weak layers as described.

Their results coincide well with the experimental results by Rask and Sørensen (2012) and they have found a model for how to design the cohesive properties to obtain a structure with optimal fracture resistance. Parameters that are manageable in a production process are the ratio of the cohesive properties of the different crack planes and the distance between the them.

The part that I would like to discuss concerns an estimation of an upper bound of the enhancement of the fracture toughness. The derived theoretical model is based on the J integral taken along a path that ensures path independence. Two different paths are evaluated and compared. Along a remote path the J-value is given as a function of external load and deformation. The structural stiffness is reduced as the crack advances in the direction of the primary crack. In the linear elastic case the J-value is half of the work done by the external load during a unit of crack growth. In an evaluation taken along a local path, J receive contributions from the primary crack tip and the two crack tips of the secondary crack. All three tips are supposed to move a unit of length in the direction of the extending primary crack.

As observed by the authors the secondary crack does not contribute to the energy release rate while what is dissipated at the propagating foremost crack tip is to the same amount produced at the healing trailing crack tip. Both crack tips propagate in the same direction so that the crack length does not change.

An observation from the experimental study was that all crack tips have different growth rates and especially the trailing tip of the secondary crack was found to be stationary. Therefore the contribution from that crack tip to the local energy release rate is annulated which leaves less available to the primary crack. To me this seems right. However, when the two remaining advancing crack tips grow does not the respective contributions to J have to be reassessed to reflect their different growth rates? If we assume that the secondary crack grow faster than the primary crack then the enhancing effect is underestimated by the J-integral. Upper bound or lower bound - I can't decide. I would say that it is a fair estimate of where the fracture resistance will end up.

In conjunction with the evaluation of the work done by the external load during a ”unit of crack growth” it seems to be an intricate problem to correlate the unit of crack growth with the different crack tip speeds. Some kind of average perhaps.

Any contribution to the blog is gratefully acknowledged.

Per Ståhle

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## Comments

## Per, thanks for raising this

Per, thanks for raising this interesting discussion.

Before I read this Blog entry I did not consider the issue of cracks propagating in different rates.

I consider the J integral as a path-independent integral. For a given situation, the cohesive zones have developed tractions. I treat the tractions from the cohesive laws as any external applied tractions - the tractions that exist at the given situation. The previous history of the cohesive tractions does not matter. Now by path-independence, the J value obtained by evaluating J along the external boundaries must be equal to J evaluated locally around the cohesive zones. It turns out that the left-hand side crack tip of the secondary crack contributes negatively to the local J integral. I find it hard to give a physical interpretation of this - hopefully the discussion in this Blog can help clarifying this.

First I want to give a clarifying comment: The problems are actually a large-scale bridging problem with a crack tip with a given fracture energy and a bridging zone in the crack wake. Thus, in the following I associate the "crack tip" with the very damage front.

And yes, the three crack tips propagate at different rates - both in the experiments and in the simulations. Actually, the left hand side crack of the secondary crack propagates in the left hand direction (the negative x1 direction), i.e. opposite to propagation direction of the primary crack and the right hand crack tip of the secondary crack (they propagate in the positive x1 direction). Using J as a path-independent integral only, I do not think it is a problem that the crack propagate at different rates. We can analyse any situation with the J integral irrespective of whether the cracks propagate in different rates. We just use the tractions that exist at the given situation.

Per, as I see it, you are considering the J integral as the potential energy release rate. In this case, I understand that it is difficult to associate the potential energy change to cracks growing at different rates.

In my opinion, Per, your interpretation holds true when you have a single crack with a well-defined crack tip (small scale fracture process zone, e.g. under LEFM conditions, i.e. when the fracture process zone is so small that it is embedded in a K-dominant region and the fracture process zone does not need to be modelled to generate the crack tip stress field, the K-field, correctly in a finite element model). I suppose that you can use this interpretation also when multiple cracks are propagating at the same rate (I think there is a few example of this in Tada's hand book).

I am not sure that your interpretation holds for large-scale fracture process zones problems (large-scale bridging or large-scale cohesive zones). For large-scale cohesive zone problems we have to include the cohesive law and cohesive zone to model the problem correctly, since the active cohesive zone will typically evolve with increasing applied load - unlike for LEFM problems where we do not even need to model the fracture process zone to generate a K-dominant region.

## The J-integral for multiple cracks

Dear Bent,

First I want to say again that I enjoyed reading your paper and the struggle you had with the secondary crack and the vanishing contribution to the J-integral.

The J-integral computes the energy release rate of whatever is enclosed if it/they should move a unit increment in the x-direction.

In your case it means, that the primary crack tip and the right secondary crack tip, require energy for growth, i.e., the cohesive energy density times the amount of crack growth. The left secondary crack tip is healing which releases the same amount of energy as is consumed at the right crack tip.

The nice parts are that 1) nothing happens to the stress distribution in the region with the crack tips and, 2) the work done by the external bending moment acting at the beam end is easily assessed by recognising that the only essential thing that happened is that the beam got extended with the short advance of all crack tips in the x-direction. All this is nicely captured by the J-integral as your analysis shows. The contribution from the secondary crack vanishes which is the expected result, though.

According to your observation the secondary crack grows at both ends. The assumption that one of the cohesive zones does not open will result in stress singularity and the contribution to the J-integral via a loop around the crack tip will probably be more or less the same negative value as before. As earlier the J-integral is still giving the energy released for the three crack tips moving a unit length in the x-direction.

As I see it, the result improves when you remove the contribution from the crack tip that is growing in the negative x-direction, but still this is not fully correct. If all crack tips are growing, the energy release rates are possibly close to the cohesive energy densities times the respective crack growth rates that you observed. The reason why I suggest this is that it will be correct for small scale bridging zones. How good it is with the large process zones that you have, I don’t know.

Another obstacle is the connection to the result for the remote path J-integral that also would give the energy released for all crack tips moving the same short unit distance in the x-direction. The calculation is based on bending of a beam that becomes longer because of the crowing crack. The ”crack growth rate” becomes essential. Should it be taken as the average measured in the x-direction even though all cracks are not moving in the positive x-direction, or maybe the average absolute value, or something else?

Per