Hello,

I am trying to find out the parameters that can be obtained from XFEM results. By analyzing the XFEM results, what are all possible parameters that can be calculated? I am especially interested in crack propagation of bimaterial interfaces. Any references addressing the same will be helpfull. Thanks.

Prathyush

dear all;

I need a code matlab for this topic CZM+XFEM, any help

Dear Julian,

When we measured the cohesive (or interfacial) strengths, we used quite large specimens such as width W=20
mm as suggested by ASTM standards. Indeed, the cohesive zone is quite small in front of a crack. Do you think the intrinsic cohesive strength should be larger than the measured one? Thansk for any comments and we plan to conduct some experiments to show the size effect.

Roy

Hello,

thanks you for the post. This is very good topic, however I am not big fun of this technique when it is applied for homogenous materials. However form my experience it gives great results when its applied for materials with microstructue discretised directly,

For more details pleas look here,

Numerical multiscale solution strategy for fracturing heterogeneous materials

http://dx.doi.org/10.1016/j.bbr.2011.03.031

Hello all,

I am beginning to explore various methods for modeling crack propagation in a bimaterial interface with no precrack.

In my limited search I have not come across a case where XFEM is used to model a bimaterial interface with no interface. Please let me know if you know any examples/publications on this.

Thanks,

Subramani

For Beginners in X-FEM ::

I did a small project on X-FEM under the guidance of Prof. Nicolas MOES & Prof. Nicolas Chevaugeon.

CLICK HERE FOR THE PROJECT REPORT

It may help the beginners in this area.

Regards,

Dibakar Datta

BROWN UNIVERSITY

www.dibakardatta.com

Dear Suku:

Thank you very much for your follow-up questions. I will do my best to reply to all of them, though altering a little bit the ordering for the sake of clarity.

To the best of my knowledge, the issue of convergence of cohesive models for problems with unknown crack path/paths is still an open problem. I think Joe’s work could be applied to numerically assess the convergence of this particular kind of nonlinear systems. However, even with a converged solution there is no guarantee that it has converged to the right one. As you suggest, performing numerical tests is the only thing we can do for now, and comparing converged solutions to benchmark experimental cases is probably the best way to go.

The fact that cohesive zone modeling provides a limited number of possible crack paths is, from my own perspective, the Achilles’ heel of the cohesive element approach. However, there are not many other options when dealing with problems involving fragmentation or pervasive fracture. The idea of this post was to summarize what people have done (and still are trying to do) to alleviate the undesired consequences of this limitation. For example, your suggestion of using adaptive insertion of nodes with a constraint of minimum distance to existing nodes, is exactly what maximum Poisson sampling does. In this case, saturation of the path deviation ratio to a value larger than one (~1.04 for most meshes) will still occur. That is, there is only so much you can do through this kind of refinement.

Cheers,

Julian

Julian,

As a follow-up to what is being discussed, couple of questions. Can convergence in cohesive simulations only be assessed in the statistical sense (eg., Joe's work)? If quasi-random seed points/nodes are inserted (with some max spacing h between nodes as an input), then in the limit that this spacing h->0, would the crack path converge and in what sense? Can numerical tests be done to come-up with a converged solution to establish a good benchmark? It would seem that the number of potential cohesive surfaces (facets of elements) would likely also dictate the crack path, since at every "step" only a finite number of paths are permissible.

Dear Ajit,

Thank you very much for your comments. I am glad you enjoyed the post! Let me try to express my thoughts on the very interesting points you raised:

1- People working on this area usually look only at the mean value of the path deviation ratio for a given mesh. While useful for convergence analysis, all information on isotropy is lost during the averaging. This is why I believe the polar plots shown above provide a better picture of the structure of the mesh. That is, the polar plot of the path deviation ratio characterizes the structure of the mesh. You can extract quantitative information from it, the most relevant ones being the mean value and the standard deviation.
2- I am not very familiar with the field of stereology but your points are persuading me to do some literature research in the area to see what can be applied to this field. Thanks!
3- It is a common practice in computer graphics to look at the Fourier spectrum of a given mesh. This provides useful information about the isotropy of the node distribution. However, this does not provide the full picture for our problem since the way nodes are connected significantly alters the behavior of the path deviation ratio. For example, imagine nodes distributed on a rectangular array. They could be connected in various patterns: square mesh, right triangles with diagonals going from up-left to down-right, right triangles with diagonals going from down-left to up-right, etc. For each of these meshes the polar plot of the path deviation ratio will look different while the power spectrum would be exactly the same.

I hope this clarifies your points. Again, thank you very much for your thoughtful comments!

Julian

Hi Julian,

Interesting topic and a clear presentation!

Also, I enjoyed your answer to Arash's excellent query above; especially relevant here are these excerpts: "... I would first make sure that the mesh has no structure. If the mesh is structured it will certainly provide preferential (low
energy) directions for crack propagation, thus generating a bias in the
solution." and "the necessity for lack of structure (randomness) makes the problem a statistical one."

Ok.

1.

Do you know if anyone has defined any statistical parameters with the aim of measuring the structure of a given mesh? And, to a bit deeper, still: what does the term "structure of a mesh," really mean? What kind of quantitative properties or attributes?

2.

Here, I have a feeling that researches involving stereology might have at least some very nice and relevant clues if not already well worked out or sophisticated answers.

Now, speaking about myself, by now, it's been almost two decades since I stopped working in that field. As a matter of fact, I seem to have forgotten all of my stereology. However, back in the early 1990s, I was a PhD student at the University of Alabama at Birmingham, and I do remember my guide Prof. B. R. Patterson showing me an examiner's copy of a PhD thesis written by a student of Prof. Arun Gokhale's; this thesis (at GATech) was concerned, if I recall it right, with stereological characterization of the digitally captured microscopic features of metals and ceramics, perhaps also including the grain boundary network.

Coming back to the present issue of quantitatively characterizing crack-paths, I think stereologists might have worked on it, even if, to them, "mesh" must have meant in the first instance something like: "a network of grain boundaries."

Thus, I very definitively think that there must be valuable inputs to this question coming from stereology; and so, it would be wonderful if someone could look up the stereological literature, or discuss it with the stereology researchers, and provide the computational engineering community with some kind of a guidance on defining quantitative parameters to measure the structure of a mesh. Also, other things like characterizing the geometry of the crack path surface, and the geometry of its propogation in simulations.

3.

Another point. Apart from stereology, can there be any other ways to measure the structure of the geometric entities like the computational mechanics mesh?

Perhaps, based on the Fourier analysis? Here, I (very vaguely) recall of the Fourier analysis being mentioned during discussions on the last year's Chemistry Nobel (Prof. Schechtman's work on quasicrystals). The Fourier analysis might not yield as detailed---i.e. acute---parameters to capture the various fine aspects concerning the geometrical structure of a mesh. However, given that FFT is computationally relatively fast, with excellent libraries (FFTW) now already available, who knows, characterizations based on the Fourier analysis might be of help to some extent.

In contrast, the measures based on stereology seem to be all: definitely relevant, perhaps already worked out to a fair degree of completeness/maturity, and practically relevant even for issues and concerns from computational mechanics/engineering.

My 2 cents.

--Ajit

- - - - -
[E&OE]

Dear Arash:

Thank you very much for your comments. Let me give you my understanding on this:

In general, for problems with no known crack path, I would first make sure that the mesh has no structure. If the mesh is structured it will certainly provide preferential (low energy) directions for crack propagation, thus generating a bias in the solution. It is very important to have this aspect in mind when using commercial software packages, since they generally generate meshes with some sort of structure as a result of their meshing technique (advancing front, etc.). For this purpose, K-means, blue-noise, or maximal Poisson sampling meshes would do the job.

The second aspect to take care of is the convergence issue. Meshes have an implicit “roughness” which in general decreases as the mesh is refined, as seen in Figure 3a. Thus, a first step would be to use adaptive refinement near the crack tip (or tips) following some criterion. The effect of this refinement would be to decrease the path deviation ratio locally. The work by Parks et al. (Parks, Paulino, Celes, and Espinha, IJNME, 2011) is a good example of this approach. However, since this “roughness” saturates about 4% error for most meshes, there is only so much you can do through mesh refinement. That is the point where I think Conjugate-Direction meshes have great potential, but that remains to be seen.

That being said, the necessity for lack of structure (randomness) makes the problem a statistical one, so I would not be happy by just looking at one specific mesh for a given problem. I think there is a necessity to look at multiple realizations of the same problem and look at the convergence in a statistical way. This is, in fact, what you are suggesting. You can read, for example, Joe Bishop’s work in this area (Bishop and Strack, IJNME, 2011).

I hope this clarifies your question.

Cheers,

Julian

Dear Julian:

Thank you for the excellent post. So, for a problem with a complicated geometry and no known solution (not knowing where the crack will go) what would be a practical solution? Trying a few different meshes and see if there are any agreements between the predicted paths?

Regards,
Arash