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Ph.D. positions

Dear fellows,

crack growth by XFEM (Cyrille Dunant, Ph.D. student, EPFL, IMX) working with us on XFEM development (xfem++) a very generic finite element library
A short note to let you know that I have now three Ph.D. positions available to fill between now and March 2008 (sonner=better) (Students from the EU only -- fully funded).

You can contact me for more information:

stephane . bordas @ gmail . com

The topics revolve around fracture mechanics and the extended finite element method and meshfree methods with applications to composite materials (including experiments) solder joint reliability (funded by Bosch) and fault propagation in reservoirs (funded by Schlumberger and EPSRC). 

They are based in the United Kingdon, Glasgow (Scotland, which is a great place to live).

The students will be integrated in a very dynamic team of young assistant/associate/full professors, including much staff with a lot of experience. 

A dozen Ph.D. students will be in the group at the start time + postdocs of high quality.

 Glasgow University was voted University of the year in Scotland by the Sunday Times (2007).

The third one will come soon, but just write me for details.  

 Best, from Glasgow,



Dear Stephane,

What an eye candy it is! I feel like doing one more PhD now!! (That is, even if I am barely managing finishing my current PhD---which, BTW, already happens to be my second attempt at a PhD!!)

But, more seriously, this .GIF animation runs too fast even on a Pentium III 933 MHz, and the picture size is too small. So, there is hardly any point opening the file in some image processing software and isolating the individual frames. So, could you please post, say on your Web site, the individual "frames" of this sequence, each with a higher resolution (i.e. with a bigger frame size)? The thing is interesting...

Is the second phase deliberately off-center? What are the assumptions related to the interface strength? Can you model, say, hundreds of second phase particles of arbitrary shapes and sizes---say like SiC particles in alumina matrix? I (and many others, of course!) would be interested in knowing how crack deflects and propagates through such a composite (i.e. one wherein both the matrix and the second phase are linear elastic, to begin with). Would like to know if you have already modeled such a thing.



Dear Ajit,

Thanks, first, good luck with your Ph.D. I know that it can seem hard at times, but you will eventually find the reward at the end of the tunnel.

Concerning the animation, you are right, they are not very clear. I will post each frame on my website. I also don't like animations too much, but they tend to attract the attention. I am glad I was successful at doing this, at least.

So. Yes, we have modelled the growth of several cracks in a matrix full of inclusions, although they were not made of the material you are mentioning.  For one cracks, things are relatively easy (even if the crack propagation criterion can be harder to calculate if you use domain integrals (for linear elasticity) and that this domain is intersected by other cracks or material interfaces (jumps in the integrand). 

If you're interested, and have possible applications of this, we are developing this code with Cyrille Dunant at EPFL, IMX, who will probably be happy to talk more about it.

 Just let me know,


PS: If you know good candidates for the positions above..... 

Dr Stephane Bordas

Dear Stephane,

Thanks for your good wishes for my PhD.

I've let a few people in India know about these studentships... However, I must let you know that most any prospective student from India would first and foremost ask if there is a scholarship or financial support available or not! So, it will help if you could please clarify what the funding situation is for the students coming from one of the non-EU countries.

About applications. I can always think of applications. But that's "applying" in thin air. As you can easily guess, my proposed applications wouldn't at all carry the weight of actual funding.

But I would be happy to know what ideas your group members/collaborators, say, Cyrille, are working on, on composites other than the fiber-laminated ones...

In the past, I had toyed around with some ideas related to stereological (quantitative) measurements of crack paths and their computer simulation... Usually, people just want to predict toughness of composites. But stereological parameters also have a value, from both microstructural design and processing (i.e. manufacturing) points of view. (After all, FGM movement began precisely because they couldn't eliminate gradients in the second phase during manufacturing anyways!)

Also, I do not know if XFEM could possibly handle local variations in the interfacial bond strengths at different points on the surface of the same embedded particle. It would be great if you could please clarify this point.

Hi Ajit,

Yes, I can also think of lots of applications! :)

Anyway. XFEM is just an approximation technique, we would need to discuss the details to see if it can handle the local variations you are mentioning. But in principle, I think it could.

I am very unfamiliar with the discussion you started in your post about composites, FGM, stereological, etc. any pointers to releveant researchers in this area?

stpehane . bordas @ gmail . com  

Dr Stephane Bordas

Dear Stephane,

1. It would be very desirable to have a computational method that could effectively handle local variations in interfacial properties for stress analysis. Past FEM models have not fared sufficiently well at the task.

2. About researchers in the various fields:

Ceramic composites: Why, Zhigang himself has done a lot of work in this area in the past.

FGM is short for Functionally Graded Materials. These materials have systematically varying properties arising due to local variations in the local population density of the second phase. This is an on-going research topic but fairly old (dates to 1980s). Any good Internet search will throw up a lot of useful sites.

Stereology: I have posted a note here.

The basic reason I mentioned the aside about FGM was to highlight the idea that it is not enough to model a microstructure as a neat spherical inclusion in a uniform matrix and predict its toughness or simulate the crack deflection due to a single inclusion. Actual world is messy and as computational methods become more powerful, they ought to take on the challenges of more messy details. (Otherwise, Eshelby's analysis of inclusion is good enough---there is no need to conduct computer simulation as such.) The SiC-alumina system is the most widely studied and, from solid mechanics viewpoint, the simplest composite system. Both the constituents here are linear elastic; there are no phase transformation issues; interface bonding tends to be fairly uniform. FGMs would be, roughly speaking, at the next level of complication. The still next level would be transformation toughened ceramics and mechanisms like micro-cracking, crack-branching, etc., which I didn't mention. I thought XFEM might be powerful enough to handle crack-paths because I saw your simulation of hundreds of cracks. The reason for adding the comment about stereology is that it has proved useful for experimental characterization of the parameters coming from the local kind of theories rather neatly, in the case of vector fields. The same should hold true for the tensor fields of stresses and strains. Crack deflection is a predominantly local issue, after all.


  I found your comment regarding the need for a method to handle local variations in interfacial properties interesting.  At first glance, I don't see a problem implementing such a constitutive law with the X-FEM.  

  What surprises me is the notion that past FEM models have not been sufficient.  I haven't looked at such a problem myself, but the only challenge I see is one of resolution.  

  Perhaps there is something I'm missing however.  What's the main issue in your experience?



Dear John,

About FEM of fracture in composites. I was talking not so much in reference to the experience of FEM models that I built, but more through my observations of the kind of studies that have been typically reported in the literature.

It is true that, as you say, ultimately, it all boils down to resolution---mainly because, FEM can be proved to approach the continuum description in the limit of ever finer element size.

But I think that the past models can be more directly (and better) described as being plain *crude.*

Here's a list of concerns I have:

(i) Despite the advertisements of being able to handle arbitrary geometry, FEM models either (a) carry plane facets in place of curved surfaces, or (b) if they at all model interfaces by curved surfaces, they soon run into the problems of non-matching surfaces upon deformation, and, especially, after fracture.

Note, curved and bumpy surfaces (like the orange peel surface) is way too much to ask for. None is even thinking of doing it anytime soon. Yet, it may matter at the microstructural level.

(ii) In a model of, say, 100 second-phase particles, to keep the FEM study to reasonable proportions, people end up modeling the entire surface of each inclusion by, say, just about 20-50 patches (per particle). This automatically sets the upper limit as to how fine certain processes such as debonding can be at all represented in that FEM model.

Again, it's a matter of resolution, but what I want to emphasize here is that the qualitative nature of the results may perhaps undergo a change as the mesh gets finer. That is to say, the coarse level mesh may not be coarsening the local changes very well.

(iii) The most serious limitation of FEM, esp. relevant to interfacial studies, is the following.

FEM, in any form of the technique, would *necessarily* predict normal stresses for any *free* surface. This limitation arises due to the functional/variational/WR formulation of FEM---i.e. the global nature of its very formulation.

The word "global" here does not refer to the global vs. local element matrix. Here, it is to be taken in the sense: averaged over the entire element, and hence, in principle unable to distinguish a surface of that element from its interior.

Therefore, for interfacial studies---where stresses *at* and near surfaces are most important---the technique carries a huge *fundamental* limitation.

(iv) Use of higher-order elements would almost invariably lead to displacement incompatibilities. Stated in plain language: Local tiny "fractures" are predicted at any small level of stress/strain.

Again, this is a minor issue in the usual structural and machine design applications of FEM, but a major concern when the purpose is modeling the selection of crack paths near interfaces in a composite.


Personally, I think the issue no. (iii) is the most serious objection. However, all other factors also are the difficulties that cannot be simply ignored in the context of fracture in composites.


Typically, when people say that FEM has been successful in modeling composites, they usually refer to studies wherein there are no local processes like interfacial debonding, micro-cracking, crack branching (one crack leading to several branches), and no modeling for the preferential selection of crack paths. So, the past successes mostly refer to things like predicting, say, the thermal mis-match stresses in the static case, that's all. Often, even homogenization hypothesis is invoked (at various levels, from constituent phases to microcracking) to let FEM at all tackle the problem.

Of course, my knowledge is a bit dated. I would be happy to know of advances covering the poorly dealt issues I mention above.

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