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# Griffith controversy

Using the Griffith energy method for analysis of cavitation under hydrostatic tension we conclude that the critical tension tends to infinity when the cavity radius approaches zero (IJSS, 2006, doi: 10.1016/j.ijsolstr.2006.12.022). The conclusion is physically meaningless, of course. Moreover, if we assume that the failure process occurs at the edge of the cavity then the critical tension should be length-independent for small but finite cavities while the Griffith analysis always exhibits length-dependence. The main Griffith idea - introduction of the surface energy - is controversial because it sets up the characteristic length, say, surface energy over volume energy. By no means is this approach in peace with the length-independent classical continuum mechanics.

The Griffith idea was a breakthrough in the thirties of the past century because the theory of linear elasticity failed to treat fracture and no other methods were available at the time. Since then, however, the formalism of nonlinear continuum mechanics has been developed together with the powerful computer techniques which allow for alternative and more consistent approaches to the failure description. As an example of such an approach I describe (IJSS, 2006, doi: 10.1016/j.ijsolstr.2006.12.022) a possible constitutive model of nonlinear elasticity where the material softening is enforced and controlled by a new material constant, which is somewhat analogous to the 'critical energy release rate'. The softening elasticity approach is length-independent and it gives physically reasonable results in the cavitation problem.

The simplicity makes the Griffith approach enormously attractive. Does simple mean correct? Should not the Griffith approach be revisited and the bounds of its applicability established?

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## Comments

## Seems interesting

Dear Konstantin,

I'd like to have a look at your paper. I can't reach your paper via the link you provided. If possible, would you post your reprint here?

Thanks,

Jae-Hyun

## Indeed Interesting

I would like to take look at your paper, which describes the constitutive law with material softening. Please post it here if you can.

Thanks

Nitesh

## Please, try the attachment

Please, try the attachment to the main post.

## size-independent or dependent?

Hi Kosta:

I don't really agree with your point in this paper. First, the conclusion from the Griffith energy method, i.e., the critical tension tends to infinity when the cavity radius approaches zero is not necessarily meaningless. Without any defects (cavity or crack), the material does not fail. The failure analysis (Griffith and yours) is based on the assumption that defects pre-exist. Larger defects give lower failure stress, and smaller defects give higher failure stress. I don't see why this critical stress has to be length independent. In fact, one of the reasons that Griffith approach is so successful is the prediction for size-dependent fracture strength: thin glass fibers fail at a much higher stress than thick glass rod. This is indeed one of the puzzles Griffith wanted to resovlve at his time.

Second, you criticized the introduction of a characteristic length due to surface energy. I don't see what is wrong with it. Yes, it is not in peace with the length-independent classical continuum mechanics. So what? The physical world is length dependent. Size matters!

Finally, the Griffith approach has its base in the principle of thermodynamics. It applies to cracks, voids, and many other problems. It is simple (to some of us), but more importantly, it has a solid foundation. So far I don't see any controversy.

RH

## Here are the problems with Griffith

1. If the failure process is controlled by stresses/strains at the edge of the void then the critical load should not depend on the void size for small voids because the stresses/strains at the edge of the void do not depend on the void size for small voids. This simple physical notion is not in peace with Griffith.

2. In my opinion, the Griffith prediction of the unbounded increase of the critical failure load with the decrease of the defect size is unphysical.

3. Griffith solves the problem in two steps. He finds stresses/strains at the first step and considers the failure criterion at the second step. Griffith is inconsistent because he ignores the surface energy at the first step and he includes the surface energy at the second step. If the surface energy is important it should be a part of stress analysis - the first step. If the surface energy is not important in stress analysis why should we include it in the failure criterion?

## Stress analysis, Griffith's theory, and the related points

I have a few points to make--some very obvious.

1. Size does matter--Griffith or no Griffith.

Consider the following simplest example. Take a plate having its center at the origin. Apply compressive loading via two point-forces, say, -100j at the point (0,10) and +100j N at the point (0,-10), where 'j' denotes the unit vector along the y-axis.

(a) Compute/Calculate the stress field for an infinite plate.

(b) Then, assume finite dimensions for the plate, say, 100 units square, and compute the solution again.

The stress fields in situations (a) and (b) are inherently different, whether:

(i) the loading is singular or not (i.e. involves point forces or not),

(ii) we are able to give analytic solutions for the stress field or not and

(iii) our numerical solution method gives proper results at the free boundaries, surfaces and edges, or not.

The stress field in (b) becomes different from that in (a) because:

(i) normal stresses can only be zero at the free surfaces/edges, and

(ii) the effect of each boundary theoretically reaches to infinite lengths.

Note, the predicted difference in (a) and (b) is exclusively based on the premises of the classical stress analysis theory.

As an aside, also note, St. Venant's principle only states what happens in the limiting scenario. But there is also an opposite aspect to it. Domain connectivity necessarily implies a modulation of the field everywhere within the domain.

Further also note, FEM is inherently wrong inasmuch as it predicts nonzero normal stresses at free surfaces. Thus, any FE analysis for (b) would have to be taken with a pinch of salt.

2. About the unboundedness increase in the critical failure stress with the decreasing defect size.

But why isolate Griffith alone here?

Why not include the whole field of elasticity? (Indeed, also all other branches of stress analysis?) After all, the same criticism would apply equally well to the simple linear stress-strain relationship too, out of the reason that this line terminates so suddenly at the yield/failure point.

Griffith's specific insight is to acknowledge the existence of the surface energy--something that didn't occur to any mechanician before him because apparently they all thought that only liquids had surface energy, not solids. To them, creation of a crack surface would take nothing, provided enough of a stress was created. To correct this misconception is Griffith's primary achievement. The fact that he could also quantitatively predict the overall failure loads for glass fibers is almost secondary in importance. The fact that the quantitative model he employed is essentially hybrid--Inglis' stress analysis thrown awkwarldy together with energetics--is of almost no consequence or significance in comparison.

Griffith's basic insight has by now been well-justified by those QM-based descriptions. We have already seen ample simulations that our "intuitions" too have become better developed. Really, wouldn't one expect that the ionic cores got ever so slightly shifted from their regular positions and that the electron cloud suffered some local density changes, whenever a free boundary/surface was introduced into an otherwise infinite lattice? Isn't this what you would "naturally" expect?

From another point of view, this issue actually is a "killer-app" for the QM-based theories of solid state physics. They should seize this opportunity to compute the surface energy due to the finite size of the stressed objects (I mean, in case they haven't, already!) and develop further along these lines.

3. Overall, Volokh's basic point seems to be that the very concept of stress at a point does not include the effects due to surface. To remedy this, apparently, he would like to distribute the effects due to the finitude or surfaces right inside each and every infinitesimal element all through the domain volume.

Now, by itself, this can be quite valuable. In fact, this is what many approaches such as the continuum damage mechanics have attempted to do in the past. Research proposals like these are welcome as descriptions of certain *particular* type of material behavior. Volokh's particular theory in the above paper is valuable from this angle.

But when this or such schemes are stretched too far and proposed as fundamental descriptions of the mechanics of solids, then the proposals must face some tough questions: Why include only the cavitation-related phenomena--say, as in the quasi-static contexts? Why not include impact-related behavior right in that infinitesimal element? Can you include the entire range for such behavior? How about the shock-related behavior? And the ductile-to-brittle transition with cooler temperature? The fatigue-related behavior? And, how about hysteresis and dissipation--in each and every one of the aforementioned phenomena? ... So on and so forth...

If the very definition of stress attempted to incorporate all such effects, the theory would become hopelessly complicated. Proper epistemology includes paying respect to the crow-principle. For further details, please see the American philosopher Ayn Rand's book: "An Introduction to the Objectivist Epistemology." Incidentally, I first read this book in 1981, and so, in a sense, there is nothing new to this point either! Yet, I think crow-epistemology is a relevant point here. Fundamentally, that's only how one could say that the continuum theoretical definition of stress (whether Cauchy's or Kirchhoff and Piola's) ought to be kept intact.

## Thanks

Thanks, though I did not understand much in your post. Probably, you can be more specific about my very specific points.

Kind regards,

Kosta