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Journal club theme of Aug. 15, 2008: Variational formulations in fracture mechanics

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.

 

Comments

Cai Wei's picture

Thanks for bringing up this classical and interesting problem.  While the paper by Bourdin is quite long, I will make an effort to read it and then come back with questions.  Right now I am thinking perhaps some of the issues you raised can be addressed by atomistic simulations, as long as the crack propagation speed is large enough so that the process is within the time scale limit of Molecular Dynamics simulations.

N. Sukumar's picture

Adrian,

Definitely a timely topic, and thanks for sharing your perspectives on the authors' work that does aid the reader. The authors  have proposed a powerful and elegant approach to fracture; I had attended a talk by one of the authors in the past couple of years and the crack growth capabilities that they demonstrated (with comparisons to experiments) appear to be unique and potentially unmatched by other existing numerical techniques.  There is some level of comfort in seeing a variational formulation as the starting point, where there is room to add enhacements withthin the same setting to better capture crack physics (thermodynamical and global vs. local minimizers as you allude to).  Will read the paper, but here are a few first impressions/questions:

  1. Extremely refined meshes are used (typically regular), but is the crack path (energy consumption)  affected by the choice  of the numerical discretization and is convergence readily demonstrable?
  2. Possibly the potential and challenge might be in the realization of numerical (finite-element like) implementations in a finite-dimensional setting that can inherit the properties of the continuous formulation?
  3. Is there an underlying stationary principle at work for fast crack growth (dynamic fracture)?
  4. A stochastic variational formulation (thought easier said than done) might be relevant given the non-deterministic nature of the physical phenomena in question.

Suku,

These are my impressions. 

  1. Extremely refined meshes are used (typically regular), but is the
    crack path (energy consumption)  affected by the choice  of the
    numerical discretization and is convergence readily demonstrable?

Yes, meshes need to be fine. The convergence is already demonstrable is some circumstances. The key condition is that the exact solution sought is a global minimum of the potential energy, and that the algorithm used for minimization should be able to find the global minimum for each discrete case. This can sometimes be guaranteed, but not always (most often this is not possible). However, in those situations, discrete solutions converge to the exact ones as both the mesh size and the parameters used to regularize the functional go simultaneously to zero in some simple but not arbitrary way. 

 Even though convergence cannot yet always be guaranteed, the fact that it can be proved to converge in some circumstances already sheds quite a bit of light on how algorithms need to be designed. 

 I must also mention that this is not the only method for which convergence can be theoretically guaranteed.  The work of M. Negri is also interesting in this regard:

 Negri M., "A discontinuous finite element approximation of free discontinuity problems"

 Negri M.,"Convergence analysis for a smeared crack approach in brittle fracture"

There are others also cited in the paper.  The problem with all of these approaches, is that they rely on seeking global minimizers, which is an arduous task for these types of problems. So effectively, it may not be possible to find the discrete solutions that do converge. Since as mentioned earlier, global minimization is not a physically sound assumption in many circumstances, the emphasis now should lie in refining the understanding of the continuous problem, instead of finding ways to obtain discrete global minimizers. Of course, this is my perspective. 

2.  Possibly the potential and challenge might be in the realization of
numerical (finite-element like) implementations in a finite-dimensional
setting that can inherit the properties of the continuous formulation?

 I commented on this above.

3. Is there an underlying stationary principle at work for fast crack growth (dynamic fracture)?

Not that I know yet. 

4. A stochastic variational formulation (thought easier said than done)
might be relevant given the non-deterministic nature of the physical
phenomena in question.

Agreed, one step at a time..... 

 

Xiaodong Li's picture

This is a very interesting topic. From materials science point of view, one may like to consider crystal lattices and planes as well as existing defects in the material. I would like to see how to get MD and DFT integrated with the formula and find out the physics meaning at the samll scales. If you or other firends happen to know papers about this, please let me know. Many thanks! 

Xiaodong,

I do not  understand your question. Can you clarify it? (what is "the formula"?)

 

 

Zhigang Suo's picture

Dear Adrian: 

Thank you so much for this selection and for writing up your own thoughts on the subject.  This is the first time that I have ever heard of this line of work, and I have downloaded the paper.  The paper is not written in a style easy for me to read, but I'll find time to try.  Your post has made the work intriguing.  Now here is an obvious question for you:  In light of this new approach, what's wrong with the conventional approach?  You have tried to address this question in your original post, but the question remains in my mind after reading your post several times, and reading the comments by our friends. 

Let me make the question more explicit, so that you have a clear target.  By the conventional approach I mean that one evolves a crack by an initial value problem by the following algorithm:

  1. Start with an initial configuration of the crack.
  2. Calculate the energy release rate G of the crack.
  3. Give a time step, and calculate the increment in the length of the crack according to a kinetic model, da/dt = f(G). 
  4. Elongate the crack in a direction according to a local criterion.

Each of the above steps may be implemented in many ways.  For step 1, for example, one might use ultrasound to detect an initial crack.  If this is not possible, one tries something else, or just whines.  It's a tough problem, intractable in its generality.  For step 2, there is a huge literature in computational mechanics. There may be reasons to turn the determination of G into an experimental problem, though.  For step 3, the kinetic model can be that of stress corrosion, determined from an experiment.  Similarly, one can use the Paris law for fatigue crack growth.  For step 4, one can select the direction of the growth such that the crack maintains a path of mode I.  When the crack impinges upon an interface, and needs to either penetrate the interface or debond the interface.  As another example, the external load may change suddenly, so that the crack may make a sharp turn in response.  In both cases, one can stipulate that the crack turns according to a local criterion.

Now this conventional approach does look clumsy at times, but it is used and makes predictions that can be compared with experiments.  As a specific example, just for the fun of it several colleagues of mine applied the extended FEM to evolve a pattern of cracks in a thin film.  I have also seen other people evolve cracks in metals under cyclic loads.

Evolving a pattern of cracks is a kinetic problem, requiring a kinetic model.

So, in what respect the new approach will do better?  Many thanks again for bringing up this fascinating topic.    

Dear Zhigang,

The  answer to your question "what's wrong with the conventional approach?"  is "nothing," at least from my perspective. The authors of the paper themselves do not actually critize the conventional approach (maybe a little). 

The one point at which they do actually take a critical perspective is in the nucleation part, since as I commented above, it is unlikely that every crack nucleates from a deffect, and they argue that a cohesive view of the interface can nucleate cracks as well. 

 The theory as it stands right now, at least as presented in this manuscript, does remove some of the flexibility you mentioned in your comment:

1) The theory does not have a kinetic law. They do mention at the beginning, briefly, that in choosing Griffith model or a cohesive model, they are in fact choosing a dissipation potential that is rate-independent. Even though the case for rate-dependent potentials  has not been analyzed yet to my knowledge, I believe it will soon be analyzed as well, from some discussions I've had.

 2) In the conventional approach the direction of crack propagation may be chosen from a number of different criteria, e.g., maximum hoop stress (as you do in the paper you cited), maximum K_I, etc. As it stands, the variational approach chooses one which, among all competing crack paths, (locally) minimizes that potential energy at each time for a given loading program. For smooth crack propagation, this may be equivalent to choosing the direction of maximum energy release rate, but I do not remember seen this explicitly proved (perhaps it has been).

 3) For nonsmooth crack propagation, it also chooses a path, even in the rate-independent case. This is an extension of current approaches, as I view it, and as I mentioned in the original post, may not correlate with experiments well, or may not apply to all situations. 

As I mentioned earlier, I see this as a snapshot of not even all developments so far, and as something that is evolving towards including more realistic physical phenomena. The progress so far, if nothing else, has brought some order to some fuzzy concepts in mechanics (at least to my knowledge). For example:

1) The precise statement of how the mathematical problem of how cracks propagate and how irreversibility is incorporated into it, is quite elegant and I would say, maybe elusive for the non-mathematician.  Your description above of the algorithm to compute crack propagation is based on a time-step. What is the continuous problem you are trying to solve? I am sure you know that well, but it is worthwhile taking a look at how the variational formulation does it. Curiously, the existence of the proposed evolution (and the well-posedness) under some circumstances is in fact proved by resorting to a time-discretization, and sending the time step to zero.

2) The sense in which convergence of numerical solutions should be understood, at least in some circumstances. Numerical simulaitons of crack evolution have been around for some time in one way or another. Besides some benchmark test that compare well with experimental results, how do we know  that the algorithm will solve the equations we seek to solve in more complex circumstances? We know how to do this very well for linear elasticity, and even for some plasticity models, but the story with cracks is much less mature, and it has been jumpstarted by the variational approach. 

 Finally, I would like to state that I am in no way an advocate of the variational approach, neither an opponent. In writing the introduction to the journal club I struggled with precisely this question you asked, and this is one of the reasons why I thought a discussion in this forum would be a nice idea.

 Adrian

L. Roy Xu's picture

Dear Adrian,

This is a very interesting topic. As an experimentalist, I always try to learn new
progress on modeling of fracture and failure. One of my movies based on
previous Caltech experiments actually is relevant to this topic.  After projectile impact and stress wave propagation, a vertical mode-I crack inside a brittle polymer approached to a
weak horizontal interface (a thin line). An interface crack was induced (a
black spot) before the incident crack reached the interface. You may find more
details from our two attached papers (Xu and Rosakis, 2003 ; Wang and Xu, 2006) .
We used a strength-based criterion to predict the interfacial crack initiation.
Needleman and his co-workers employed cohesive elements to simulate the
interfacial crack propagation part. I hope someone can simulate the whole
process.  I have many crack initiation and propagation photos and data to share with you.

For more movies recorded from a high-speed camera (click here). It will
take a few minutes to access my movie site since the size of each movie is
quite large. But the movie resolution and layout from my site is much better
than the movie from YouTube. © L. R. Xu (Vanderbilt University) and A. J. Rosakis (California Institute of Technology)----- Roy

gurses's picture

Thank you for bringing up this interesting topic. I would like to point out another variational formulation of brittle fracture and its finite element implementation. This can be found in the following works where a consistent thermodynamic framework for crack propagation in an
elastic solid is presented. It is shown that both the elastic equilibrium response
as well as the local crack evolution follow in a natural format by
exploitation of a global Clausius-Planck inequality in the sense of
Coleman's method. These are more in the same line as classical works (Stumpf and Le: Variational Principles of Nonlinear Fracture Mechanics, Acta Mechanica, 83: 25–37, 1990 and Maugin and Trimarco: Pseudomomentum and Material Forces in Nonlinear Elasticity: Variational Formulations and Application to Brittle Fracture, Acta Mechanica, 94: 1–28, 1992) with a strong empahsis on configurational forces.The following works still requires an initial defect for a crack to propagate, in other words a crack initiation cannot be predicted.

A robust algorithm for configurational-force-driven brittle crack propagation with R-adaptive mesh alignment
C. Miehe, E. Gürses
International Journal for Numerical Methods in Engineering Volume 72, Issue 2, Pages127 - 155

http://dx.doi.org/10.1002/nme.1999

and

C. Miehe, E. Gürses and M. Birkle
A computational framework of configurational-force-driven brittle fracture based on incremental energy minimization
International Journal of Fracture Volume 145, Issue 4, Pages 245-259

http://dx.doi.org/10.1007/s10704-007-9078-1

 

 

A 3-D finite element implementation of these works has been done and a paper has been submitted. Since it is still  in review process, I do not post it here. However, I can refer to my PhD thesis for anyone interested.

E. Gürses

Aspects of Energy Minimization in Solid Mechanics: Evolution of Microstructures and Brittle Crack Propagation
http://elib.uni-stuttgart.de/opus/volltexte/2007/3258/pdf/thesis.pdf

Dear Adrian, 

Pradeep brought my attention to your posting, and it is good for your to initiate such discussions and debates.

The paper proposes a new variational approach to fracture problems, and it is refreshing at certain level. At least, it is an out-of-box approach. Zhigang asked a pertinent question: What is wrong with the old approach ?

If I may, I woud like to add: What is right with the old approach ?

The old approach now has an official (sexy) name: Configurational Force Mechanics, and it has become ever popular and exciting. In fact,next month, a group of elite mechanicians will gather at Kaiserslauten, Germany on another Configurational Mechanics Symposium --- that is another story.

The beauty of the configurational force mechanics is that the variational structure or variational principles that are adopted are the very principles in standard continuum mechanics or Newtonian mechanics. The configurational forces or the configurational mechanics come out as the invariant properties of the very variational principles of continuum mechanics, which we call as Conservation Laws. They provide the basic governing equations on defect motions based on the initial continuum modeling. In this sense, the configurational mechanics is consistent with the standard continuum or solid mechanics in both physics principles as well as mathematics structures. We do not change simulation model and modeling methodology in the middle of an analysis or a simulation.

Whereas the discussed approach uses different or additional variational principles to treat the fracture problem as if this is an entirely different physical phenomenon governed by entirely different or additional physical principles. I have no intention to make a judgement on the merits of the paper, but just from this perspective, the approach violates the spirit of minimum structure (both mathematical and physical) requirements of a fundamental theory. Nevertheless, I do admire both the authors and Adrian for their physical as well as mathematical sophistications. 

I may add that some of the physical principles that the paper used are indeed physical and sensible. The point is that: Can we derive them naturally from the initial continuum model ? If someone changes the initial continuum model, and subsequently he brings out some desired crack evolution laws based on the initial continuum model, or he makes the crack evolution law consistent with the initial continuum model, that will be a significant contribution, at least in my opinion.

 

 

Shaofan

 

 

 Dear Adrian:

   First I want to thank you for bringing up the topic of "variational fracture” in your journal club, and also for reporting so incisively on our work and that of our colleagues. You certainly did a superb job at explaining the basic tenet of the theory and clearly pinpointed some of its strengths and weaknesses.

I would like to take this opportunity to shed a different light on some of the issues you evoked in your thoughtful analysis, and also maybe to address some misperceptions on the part of your various correspondents.

Our main goal and drive in devising the variational approach was and remains the prediction of the crack path without the import of additional ad-hoc ingredients. I think we clearly achieve that goal in the global minimality setting, while acknowledging that such a setting assumes unrealistic foresight on the crack’s part. Locally minimizing paths, if demonstrated to exist and to be different from globally minimizing paths, might prove somewhat more realistic, but deriving a locally minimizing path  is at present a mathematical challenge. Let me remark that the lack of an adequate machinery for local minimality  is a plight which is shared by even the best established mechanical theories; think for instance of finite elasticity where global minimality of the potential energy has to be assumed for existence under general loads, and this with no more justification than in the present setting. As a matter of fact I do not know of any non-convex setting where one does not run into this conundrum.

In any case, maybe we made too much of a fuss in our work about this, and it has obscured another important feature of the work, the regularity (or lack thereof) of the crack path and the smoothness (or lack thereof) of the crack evolution along that path. Indeed, if the crack path is smooth enough and if the growth of the crack along that path is smooth then our formulation and Griffith’s are one and the same. Thus, contrary to what is being gently intimated by Shaofan, there is no new physics in what we do. We are Griffith and Griffith is us, provided Griffith makes sense. The problem is that more often than not, Griffith grinds to a halt for many reasons. In this respect there is something wrong with the conventional approach. And your readers know that, because they have to add extra features to Griffith in many situations: the principle of local symmetry for kinking, unless it is the G-max criterion, Paris law of fatigue for cyclic loading,….

We posit that some kind of local minimality criterion, plus a decision on the form of the surface energy, is all we need in all situations – at least in a quasi-static setting. Hence for example the – yet not completely rigorous -- derivation of a Paris type law, solely starting from an adequate re-scaling of the fracture problem (using a cohesive type of surface energy) in the last chapter. Similarly, we have new results (not contained in the work you reported on) about crack kinking which could put an end to the endless debate on the nature of the kinking criterion. So, if there is new physics, it is in the acceptance of unsmooth cracks sites and of possible jumps in crack lengths at certain times. Once again, this is no more shocking in my opinion than passing from differentiable solutions of an elasticity problem to solutions which live in weaker spaces like Sobolev spaces,…. It might be the case – although I strongly doubt it -- that real cracks are always smooth and that the crack length is a smooth function of time; this becomes a regularity issue. But why should it be assumed from the get-go.

I would also like to briefly discuss the issue of  numerics. The method we propose, which is an adaptation of a well-known method in image segmentation, has a definite edge over other methods such as extended finite elements. First it lets the crack be where it wants to be. Then, and here I have a slight disagreement with you, it does not require any mesh refinement. The computations  of Blaise Bourdin are performed on a fixed mesh. Of course one should avoid a mesh that has a preferred orientation. But that is about it.  The issue of finding a global minimizer of the time and space discretized problem is  indeed a difficult one (once again there is nothing new here, the same issue arises in e.g. finite elasticity). Now it may actually be the case that the numerics do better than the theory because the computed path is always along stationary points of the energy functional, which may actually be closer to what we would want if the theory permitted that.


The convergence results of the discretised approximation to the fracture evolution only apply for now to global minimizers, so that caution should prevail, although preliminary results do assert that stationary points of the approximating functional – and those are surely obtained by the proposed numerics – do converge to stationary points  of the fracture functional in the rather uninteresting 1d setting.  This is where we stand on the numerical front. Those that are interested may want to visit Blaise Bourdin’s website (www.math.lsu.edu/~bourdin/) where many numerical examples are produced.

As a final note, I would like to point out that I agree with Adrian that dynamics are needed as a selection mechanism for crack evolutions. There is some effort on that front on the part of J.J. Marigo and collaborators and also on that of B. Bourdin and C.J. Larsen. At present, the mathematics are lagging far behind.

I hope  the above remarks, which I claim sole responsibility for,  have not obscured the issues that you and the other contributors have raised. once again, I thank you and the other interveners for your interest in this topic.

 

 

 

 

Gilles Francfort, Université Paris-Nord

Dear Gilles,

First of all, let me note that I really enjoyed your above comment... All of it. It's been written beautifully.

But most of all, I was impressed by the honesty evident in this part (the part that happens to interest me the most: physics and physical interpretations of mathematical constructs).

"Our main goal and drive in devising the variational approach was and remains the prediction of the crack path without the import of additional ad-hoc ingredients. I think we clearly achieve that goal in the global minimality setting, while acknowledging that such a setting assumes unrealistic foresight on the crack’s part.
"

Now, that is what one always wanted to see being said in print---but never did find it being said so explicitly. (My apologies in advance if some variational researcher had in fact said it earlier---my reading is limited.)

-----

As a student of variational principles---of their physical bases---I always wanted to have a point or two about them clarified from someone.

However, most people seldom seem to be objective about this category of principles---usually, they are too fixated about them, or are too "devoted" to them, sometimes, to the point of blatant irrationality.

I think that any principle involving a variational formulation (whether for problems involving crack growth or optics or quantum mechanics), would necessarily have to be global in nature. Here, by global, I mean: the thing being varied must have support everywhere in the domain (in both the space and the time coordinates as the problem might require).

Based on my (limited) thinking, I fail to see how the aspect of variations could at all be integrated with certain elementary considerations like conservation and symmetry, unless the physical quantity that is varied were not to be spread out all over the domain---over at least a finite part of it. The support cannot be infinitesimal, that's the point.

Thus, true locality in the sense of a point-phenomenon seems impossible in the context of variational methods, whether someone has already elevated this observation into the form of a principle or not.

Yet, you speak of "local stationarity" etc. ... Is this (the local variational formulation) just an abstractly hypothesized possibility, or is there a something more specific to it?

For every formulation advertised as "local", a closer inspection invariably reveals that the underlying base, in fact, is global. If not directly and explicitly, then, at least, indirectly and subtly.

If so, what does the term "local variational formulation" really mean? To me it seems like an in-principle meaningless phrase. Can anyone suggest any concrete example(s) counter to my position? Any specific examples that were illustrative of this phrase (in any kind of field problems---not necessarily involving elasticity or fracture, but any other kind of field problem also)?

Any thoughts or comments? Thanks in advance.

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