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# Fracture and singularities of the mass-density gradient field

To appear in Journal of Elasticity

A continuum mechanical theory of fracture without singular fields is proposed. The primary

contribution is the rationalization of the structure of a `law of motion' for crack-tips, essentially

as a kinematical consequence and involving topological characteristics. Questions of compatibility

arising from the kinematics of the model are explored. The thermodynamic driving force

for crack-tip motion in solids of arbitrary constitution is a natural consequence of the model.

The governing equations represent a new class of pattern-forming equations.

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## Comments

## interesting theory Amit

dear Amit, interesting theory, although I am not an expert in the field to judge completely.

I have always been reluctant to accept singularities, and in the end, many theories take singularities, but then return to non-singular solutions, for example in fatigue do you know of the so-called "Theory of critical distances" which has been popularized by David Taylor but is in fact existent much before him? There is also a version of "Quantized Fracture Mechanics" by Nicola Pugno which is similar.

Perhaps your theory can be connected with this (much simpler) ideas?

## Re: Interesting theory...

Dear Michele,

Thank you for your comment and the references. I was not aware of the works by Taylor & Susmel and Pugno and Ruoff you mention, but now will be, thanks. I need a little time to look through these and try to form an opinion, at which time I will get back to you.

For me avoiding the singularity is important in so far as models for evolution of defects inevitably require objects like stress and energy density at the defect for driving force and in nonlinear theories it is very difficult if not impossible to separate out `my stress' from the defect itself from 'your stress' of other defects and boundary conditions.....if I was only interested in energies and minimizing it for results, this would not bother me so much except if global energies of finite bodies started becoming infinite, which happens for dislocations, but not for cracks or disclinations.

On the question of simplicity - I guess it is in the eye of the beholder. I have spelled out my goals in the paper, and am interested in a conceptual framework for dealing with evolving singularities in continuum mechanics that can lead to practical advances (and for me this means through robust computation). I also want to be faithful to the different kinematics related to different types of singularities and in understanding how to accommodate such in the energy. Curiously enough, this directly connects to the `technology' of gauge theories in physics, where they pull out these kinematic fields from the gauge principle. I find it much more satisfying to find such things emerge simply from trying to understand singularities, which is what the deeper message is, in my opinion, in the works of Weingarten, Volterra and Mura and DeWit. And if the principle of gauge invariance is not satisfied, so be it.....

Anyway, I am probably rambling here, so I will stop. But simplicity for me is defined as a model that

1) has generality, so it does something for you beyond an immediate problem you want to solve, preferably shows some new way of thinking (since one has to work hard at the problem anyway), i.e. does not make one miss the forest for the trees

2) you can command an electronic device with not much intelligence to execute, once given precise instructions. The precise instructions may require very detailed algorithms related to subjects like nonlinear wave prorpagation etc. which in itself takes a lot of learning.

I don't think the current state of affairs related to dealing with evolving singularities in continuum mechanics is at a stage of theoretical and algorithmic development where this claim can be made, but progress is happening.

Taylor, D. (2010). The theory of critical distances: a new perspective in fracture mechanics. Elsevier.

Taylor, D. (2008). The theory of critical distances. Engineering Fracture Mechanics, 75(7), 1696-1705.

Susmel, L., & Taylor, D. (2008). The theory of critical distances to predict static strength of notched brittle components subjected to mixed-mode loading. Engineering Fracture Mechanics, 75(3), 534-550.

Susmel, L. (2008). The theory of critical distances: a review of its applications in fatigue. Engineering Fracture Mechanics, 75(7), 1706-1724.

Cornetti, P., Pugno, N., Carpinteri, A., & Taylor, D. (2006). Finite fracture mechanics: a coupled stress and energy failure criterion. Engineering Fracture Mechanics, 73(14), 2021-2033.

Pugno, N. M., & Ruoff, R. S. (2004). Quantized fracture mechanics. Philosophical Magazine, 84(27), 2829-2845.

## no offence in "much simpler"!

dear Amit

when I said that the "critical distance" approaches are perhaps "much simpler", I wasn't meaning to say that your theory is too complicated! The critical distance approaches are very empirical, not evolved in general, and I do not think they have a solid background. Yet, they work quite well, so I am happy you say you want to have a look.

Perhaps as you mention the "peridynamics approach of Silling", you could consider that "critical distance" methods are similar: they both define what Silling calls an "horizon".

Have you made any further comparison with peridynamics? If so, you are quite close to "critical distances".

## Re: no offence in "much simpler"

Dear Michele,

I did not take offence at all (it takes a lot to get me offended in a technical discussion; hopefully never).

While I have not yet looked at the papers you mentioned (I definitely want to have a look - I like to take a look at everything I can!), I have a reasonable sense of the basic idea of what goes on in peridynamics. My model is a pde model and not nonlocal (ok this depends on what variables one is working with) and so there are some big differences in that respect..... an interesting question that comes to mind is what is the limiting form of the perdiynamic model in the limit of zero-horizon. That might be a pde model and looking at the connection of that to my model may make sense. However, in the limit model I am not sure if there can be a length scale, and my pde model certainly does have at least one intrinsic length scale which will be operative even in equilibrium solutions, so....

## "critical distance" in few words

The critical distance in few words (so that you can avoid reading the entire Taylor book!)

1) point method:- this has the standard singularity of the classical solution, but neglects it! So it simply considers a stress value at some point ahead of the crack --- of a given material constant distance

2) line method: there is some averaging of stress here: results are very similar to PM above, and perhaps this becomes "non-local".

3) area method and volume method --- the average is taken over area and volume, but again no big deal

4) quantized fracture mechanics: the average is taken of the stress intensity factor

What do you think of these?

## another quick elementary reading is this:

The Theory of Critical Distances (TCD) is a bi-parametrical approach suitable for predicting, under both static and high-cycle fatigue loading, the non-propagation of cracks by directly post-processing the linear-elastic stress fields, calculated according to continuum mechanics, acting on the material in the vicinity of the geometrical features being assessed. In other words, the TCD estimates static and high-cycle fatigue strength of cracked bodies by making use of a critical distance and a reference strength which are assumed to be material constants whose values change as the material microstructural features vary. Similarly, Gradient Mechanics postulates that the relevant stress fields in the vicinity of crack tips have to be determined by directly incorporating into the material constitutive law an intrinsic scale length. The main advantage of such a method is that stress fields become non-singular also in the presence of cracks and sharp notches. The above idea can be formalized in different ways allowing, under both static and high-cycle fatigue loading, the static and high-cycle fatigue assessment of cracked/notched components to be performed without the need for defining the position of the failure locations

a priori.The present paper investigates the existing analogies and differences between the TCD and Gradient Mechanics, the latter formalized according to the so-called Implicit Gradient Method, when such theories are used to process linear-elastic crack tip stress fields.

Askes, H., Livieri, P., Susmel, L., Taylor, D., & Tovo, R. (2013). Intrinsic material length, Theory of Critical Distances and Gradient Mechanics: analogies and differences in processing linear‐elastic crack tip stress fields.

Fatigue & Fracture of Engineering Materials & Structures,36(1), 39-55.## unless you want your cracks to move

I think we agree that for the static solutions nothing much will be different -– gradient elasticity says nothing about how to get cracks to move, if I am not mistaken…

But perhaps your main interest is in having a model where one can make crack tips move and multiple cracks interact. Interaction perhaps is partly accounted for in "critical distances", but moving cracks not.

Then, the second question is how to move your cracks. Dynamically?

## Re: unless you want cracks to move

The dynamics in the model does not have to do necessarily with inertia. For the brittle case, it is the c eqn 24_3, and it can be solved with no inertia in 24_2.

The motivation for why and how the model would move crack tips is in sections 2 and 3 (before 3.1). Also the concluding remarks section.

As for statics, yes I more or less agree with you, except I see possibilites for putting in atomistically defined inputs in the more realistic analogs of the functional dependence on of the energy on c and curlc (and even in the dependence of the 'elastic' part on c).

## but what is driving the motion, and is Griffith theory included?

dear Amit

I see the equations, but I am not able to see where the driving force is for cracks to form in the first place, to increase their stress concentration, and eventually "become unstable" as in Griffith theory. Are there discontinuous surface in the end in your model or not? Is Griffith theory recoverable as special case?

If you are talking of brittle fracture, I do not see, within elasticity, how you can neglect inertia. Crack propagation becomes classical unstable then, and a quasi-static model serves to nothing.

The case of fatigue is another story. If you mean to model crack propagation, you need to convince people that you have Paris law as special case, rather than Griffith theory.

Hope this helps.

## Re: but what is driving the crack motion....

Dear Michele,

I agree with you (and as said in the paper) that showing the connections to Griffith theory and Paris law (in the case of cyclic loading), and in general to existing fracture mechanics concepts, form major tasks of the model. I consider this first paper as a modest contribution and not a magnum opus about to change the face of the subject.

That said, by translating a particular type of dislocation model in papers with my students Xioahan Zhang and Chiqun Zhang (used also for defect propagation in nematics) to the present context, I believe the questions you ask can be well answered - with a prediction for G_c in terms of model ingredients (I base the remark on preliminary work in this context). Making a fundamental connection to driving conditions for extension under fatigue loading is obviously a more involved question of emergence, but can be attacked by the reduced model I am talking about. In any case, that is neither here nor there at this stage, so we will simply have to wait and see.

In the model, any crack evolution is accompanied by dissipation - even in brittle, nominally elastic materials - this represents the energy dissipated in bond breaking e.g., that cannot be captured by motions representable by a continuum field. In this sense, the model is never elastic, and I disagree with you that a quasi-static model serves nothing - just think of the Peierls stress model for dislocations, as one example of unstable propagation (this we have dealt with in great detail). I am not implying energy going to inertia should not strictly enter the energy budget, and it can be handled by the model...

Limit solutions of the model will give discontinuous surfaces across which no tractions are transmitted. Off the limit this is a model for notches.

I am not willing to commit to what the model will do for homogeneous nucleation of cracks (there are comments in the paper, and one needs to see). I don't think there are any good fundamental models for this at reasonable time scales....

The driving force for extension of an existing crack comes from elastic strain energy (accounting for plasticity, if included).

I don't feel the need to convince anyone of anything, but simply to work out the consequences of the model and see where that leads. But I do thank you for your comments.

## dislocation, cracks, notches,

I think if I knew more about dislocations perhaps I would understand your model better.

I loose you when you say that in one limit you should tend to the crack --- the limit on which quantity?

And also when you say " Off the limit this is a model for notches." --- what do you mean by notch? For me, notch has also no traction transmitted across the surface, for me a notch is, for example, a hole in a plate.

I suppose if we defined a common language we would make progress.

## Re: dislocation, cracks, notches,

just to fix ideas, considering the static case with c field specified, it would be the length corresponding to c_w in Figure 1a.

When c_w is not zero but small, then the model will be for a notch, and yes, this also does not transmit tractions.

Beyond what I have written down, I have/will have no further progress to report without further work.

## then, why not submit it to some journal?

I also think I have nothing more to ask without further work: without some comparison with existing knowledge, existing theories, existing experiments, I find it difficult to judge :)

Perhaps if you send it to some more mathematically-oriented journal on fracture, you may get some reviews and take it form there....

## Re: then, why not submit it to some journal?

Thank you for the advice. It has of course been submitted to a journal.

Note added (Nov. 7, 2017) - the paper will appear as a research paper in Journal of Elasticity. The accepted version of the paper has been uploaded.