This issue of the journal club addresses a topic deemed classical by most mechanicians: the onset and evolution of fracture in brittle solids.  In the last ten years or so a community of mathematicians have been analyzing the formulation of fracture mechanics models.  Although still at a very early stage, there is already a substantial body of work, which either sets in a very rigorous settings some known results, or give rise to other perhaps more surprising ones. Some of the ideas may be deemed physically unfeasible, proposed out of mathematical convenience, but progress onto more accepted ones is slowly happening. The strong mathematical basis of the theory is also impacting the understanding and formulation of numerical methods for crack propagation.
 
 
In choosing a topic for the journal club, I judged that this was a good time for mechanicians to take a look a some of these developments, and have a discussion about it. To this end, I chose only one recent paper to read, albeit a rather long one. The article was recently released as a book as well.  The reference is

Bourdin B., Francfort G. and Marigo, J., "The variational approach to fracture", Journal of Elasticity, 91, 1-3, April 2008, pp 5-148.

http://dx.doi.org/10.1007/s10659-007-9107-3

Of course, this is not the only article on the topic, but it is a recent one, written without much of the mathematical elements most others have. It also has the spirit of an overview, in the sense that it truly attempts at placing in context the author's contributions for the most, and of others to a lesser extent. In this sense, it is a rather accessible article for mechanicians with some background in analysis.

About the variational formulation:

The starting and central idea of the theory is to formulate the crack evolution in a purely elastic body as a minimum principle. The novelty lies in that, given a quasi-static loading program for a sample, the deformation field and the crack locus are those that minimize the potential energy of the system at each  "time'', among a large set of competing deformation fields and crack paths. In order to select evolution in which any decrease in the elastic energy upon cracking goes into surface energy, an  "energy balance'' condition is added.

The first controversial aspect of this formulation, widely discussed by the authors in the paper, is whether to seek stationary points or (global or local) minimizers of the functional. A section of the paper is dedicated to explore the consequences of each one of these choices in simple examples.  Global minimization often leads to unphysical results, as illustrated in the paper and known for some time (energy barriers do not exist in this case). It is mathematically convenient though; which it is clearly not a good reason to stick to it in the long term. One may argue that some sort of thermodynamic stability condition should be invoked to discard a stationarity condition that is not a local minimum, but this is not part of the theory so far.  I would personally favor adopting a local minimization condition, which the authors mention as the most promising avenue as well. The precise class of perturbations considered to test these minimizers, and what type of evolutions they can predict, are yet to be defined.

The theory extends Griffith's ideas further to encompass a wider set of phenomena. Griffith's theory does not explicitly address the propagation of the crack; it is only a stability argument. This perspective could be controversial, since different scientists may have different interpretations of what Griffith's theory precisely is. For example, in the case in which crack propagation is not smooth, there is no way to predict from Griffith the crack propagation path. The variational framework does include this case. The question of whether the chosen path is physically correct is open though; some aspects of the dynamics of the process will likely be needed here to make the case.

The rigorous analysis of some simple problems in the paper also proves enlightening, as the one commented next. The standard argument against Griffith, already made by Barenblatt a long time ago, is that infinite normal traction needed to propagate the crack. Some of the weaknesses of such assumption are also explored in the paper. Instead, if a cohesive model is adopted, the necessary normal tractions are finite, and a perhaps surprising (some expert mechanicians are probably aware of this) outcome is the possibility of obtaining distributed microcracking (damage) from exactly the same variational formulation when global minimizers are sought. No additional features are needed.

Similarly, crack initiation is possible in the variational approach with cohesive models, without the need of preexisting defects or micro-cracks. It is difficult to argue that crack propagation "always'' begins from defects, as the authors of the paper mention.  On the other hand, the variety of possible crack paths encountered for theoretically "identical'' samples is an argument in favor of the nucleation by defects; otherwise such a spread would not be expected. The extent to which a theory could be expected to predict crack nucleation is questionable, mostly because the exact location of the defects is generally not known. The question of "what if you did know....'' is still valid nevertheless, and gives some reason to pursue it.

About numerical solution of fracture problems:

The variational formulation of fracture problems provides some ideas on how to evaluate and design numerical methods for fracture mechanics. In particular, a sense of convergence can be defined. The first and foremost requirement is that the space of discrete deformation fields and crack paths should be rich enough to simultaneously approximate both the elastic field and the surface energy as the mesh is refined. It is easy to construct counterexamples in which this is not the case if cracks are allowed to propagate through element boundaries only, and the crack surface is simply computed as the area of that face. This provides a strong rationale for methods that enrich the set of admissible crack path, such as the extended finite element method.

The drawback of methods in which the crack path is explicitly represented (or perhaps the blessing), is that it is difficult to include the crack path as an optimization variable, as the variational framework suggests; crack propagation is instead generally staggered with the solution of the elastic field.

Ultimately, the explicit representation of the discontinuity should not be over-interpreted in the discrete setting. There are many ways to do that. The authors in this paper advocate a way of regularize the discontinuity and spread it over several elements. The advantage of this approach is that, in the optimization setting, crack paths are a result of a (very difficult, not always well-defined) optimization. Bifurcations and non-smooth crack growth are naturally included here.  The disadvantage is that the needed refinement of the mesh near the crack significantly increases the computational cost.

In a related work by M. Negri, the effect of the mesh on the surface energy approximation is discussed, demonstrating that meshes have to be "isotropic'' to avoid inducing a numerically anisotropic surface energy term.