iMechanica - Comments for "Discussion of fracture paper #14 - How to understand the J-integral when multiple cracks are growing at different rates"
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Comments for "Discussion of fracture paper #14 - How to understand the J-integral when multiple cracks are growing at different rates"enThe J-integral for multiple cracks
https://www.imechanica.org/comment/28471#comment-28471
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<p><em>In reply to <a href="https://www.imechanica.org/comment/28453#comment-28453">Per, thanks for raising this</a></em></p>
<div class="field field-name-comment-body field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><p><span>Dear Bent, </span></p>
<p><span>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. </span></p>
<p><span>The J-integral computes the energy release rate of whatever is enclosed if it/they should move a unit increment in the x-direction. </span></p>
<p><span>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. </span></p>
<p><span>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. </span></p>
<p><span>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. </span></p>
<p><span>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.</span></p>
<p><span>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? </span></p>
<p><span>Per</span></p>
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</ul>Sat, 23 Jul 2016 09:37:00 +0000ESIScomment 28471 at https://www.imechanica.orgPer, thanks for raising this
https://www.imechanica.org/comment/28453#comment-28453
<a id="comment-28453"></a>
<p><em>In reply to <a href="https://www.imechanica.org/node/20004">Discussion of fracture paper #14 - How to understand the J-integral when multiple cracks are growing at different rates</a></em></p>
<div class="field field-name-comment-body field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><p class="MsoNormal">Per, thanks for raising this interesting discussion.</p>
<p class="MsoNormal">Before I read this Blog entry I did not consider the issue of cracks propagating in different rates.</p>
<p class="MsoNormal"><span>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.</span></p>
<p class="MsoNormal"><span>First I want to give a clarifying comment: The problems are actually a large-scale bridging problem with a</span><span> </span><span>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.</span></p>
<p class="MsoNormal"><span>And yes, the three crack tips propagate at different rates</span><span> </span><span>- 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.</span></p>
<p class="MsoNormal">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.</p>
<p class="MsoNormal"><span>In my opinion, Per, your interpretation holds true when you have a single crack with a well-defined</span><span> </span><span>crack tip (small scale fracture process zone, e.g. under LEFM conditions, i.e. when the fracture process </span><span> </span><span>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).</span></p>
<p class="MsoNormal"><span>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.</span></p>
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</ul>Tue, 12 Jul 2016 19:24:00 +0000Bent F. Sørensencomment 28453 at https://www.imechanica.org