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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.

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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.

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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.