Muhammad this is a big question which a lot of people have tried to answer, but there is no simple answer. I would say that the distance represents the role of the microstructure in the particular failure mode being studied (e.g. fatigue crack growth, brittle fracture etc). It's a simple way of bringing in the effect of the microstructure without using some very complex micromechanical modeling. I've written something about this recently in Taylor D (2016)  On the role of microstructure in finite fracture mechanics Procedia Structural Integrity 2:1999-2005.

Muhammad this is a big question which a lot of people have tried to answer, but there is no simple answer. I would say that the distance represents the role of the microstructure in the particular failure mode being studied (e.g. fatigue crack growth, brittle fracture etc). It's a simple way of bringing in the effect of the microstructure without using some very complex micromechanical modeling. I've written something about this recently in Taylor D (2016)  On the role of microstructure in finite fracture mechanics Procedia Structural Integrity 2:1999-2005.

Hi David,

I went through your work on FFM. It is interesting how you link the length parameter to micro structure of the material. My question is more about the length scale used in non-local theories of damage mechanics. The phenomenon of damage starts much earlier before a macro/meso-crack is formed.

Question is what kind of microstructure you are considering as a source of the collective effect that you call damage.

In this sense, David is right. To be more precise: if you consider a microcracking process as damage source, the lingth scale indicates from a physical viewpoint the size of a spatial neighborhood around a point x, which contains the microcracks interacting with the one that you imagine within the material element at x.

Such an interaction is scattered in a field theory. You may measure it as a perturbation of the standard stress (there are papers by Tom Hughes in this vein) or you can consider it as a contact interaction a part, a microstress satisfying its own balance equations (there are papers of mine adopting this view). In this case you avoid the ill-posedness arising in the hardening-to-softening transition, and the mesh dependence.

In a field theory the length scale disappear at the level of kinematics and the derivation of balance equations because it is idden in the choice of considering the gradient of the damage indicator. However, it re-appears in assigning constitutive structures. A way to get them is to identify constitutive relations from a (multi)lattice scheme, as I did in a couple of papers on Comp. Meth. Appl. Mech. (2001,  38, 1963-1974; and 2004, 193, 5035-5062). In this case you can even consider the length scale as a stocastic field over the body, as I did in a paper on Prob. Eng. Mech. (2004, 19, 127-143).

Hi Paolo,

Ok. But what you have explained seems that we know the crack density distribution (or defect distribution) and we also know how do they interact. In that case yes you can measure the size of the neighborhood which influences the local point 'x'. Please correct me if I understood it wrong.

The question still remains that how can we find the size of the interacting neighborhood apriori. For e.g. if I want to 'PREDICT' the failure in a certain model with a certain size and loading condition then what will be the basis to determine the length scale for that model. What kind of tests do I need to perform to obtain a length scale that can fit 'generally' to a complex geometry and loading condition?

You know the crack (or defect) density distribution as a solution to field equations.

You imagine, e.g. on the basis of tomography analyses or even just on direct observation, an initial distribution (the one that you use in deriving from a lattice scheme constitutive equations) and find the one at the instant t furnished by your field equations. Then you compare the predicted result with what emerges from standard traction tests, and you rearrange the initial ansatz about the defect distribution.

If you take a look at the references I indicated, you may have an indea. If they are not at your disposal, you can write me by standard e-mail and I can send them you.

I cannot upload all my reprints on RG and I do not have all the preprints (I shall upload those I have, no yet uploaded, in the next days).

Our models must have predictive nature but they are only representations of our phenomenological world. In this sense they are based not only on experiments but also on our view on the phenomenon. Also, theory and experiments interact. Once you perform an experiment, you have already in mind a theoretical view of the phenomenon you are considering. In contrast you do not know what you have to measure. The question is not what kind of experiment, rather it is what kind of model and experiments you have in mind.

About non-local models, take care! When you have a gradient-type nonlocality, it works well. In contrast, when the body is infinitely extended in space, integral type space nonlocalities work also but when you apply them to describe bodies with finite size, about the boundaries, you must cut the kernel of the integrals you use and the result depends on the type of cut you decide to adopt.

As regards O. M. E. Suleiman's answer, some remarks should be added to the files 9.pdf and 10.pdf.

1) The proposal in 9.pdf has operational character. In a sense it a way to adjust computationally things rather having a fundamental physical nature. That choices have physical ground, indeed. However, what about their physical necessity? Better, alternative choices can be considered.

2) A list of such choices appears in 10.pdf. Then it is reduced to a rather traditional approach. However, two aspects deserve attention in that reference.

(a) When Jirasek mentions the possibility of considering the small strain tensor field as endowed with a "regular part" (in his words) and a "singular part", he is implicitly considering deformations as special bounded variation functions (SBV) and the strain tensor field as a measure endowed with a part absolutely continuous with respect to the volume measure (Lebesgue's one) and a singular part concentrated over a rectifiable set with codimension 1. However he does exploit further the assumption, addressing his attention on what he calls "regularized models".

(b) It is the first class in his list, i.e. continua with microstructure, precisely something that we can extract from the general model-building framework for the mechanics of complex materials, that can furnish a physically sound and well-grounded approach to the question. Something in this direction is in the papers that I sent to Muhammad, answering his e-mail.

As a marginal remark, if we would prosecute this discussion, in the end we would have material to start a paper.

Dear Osama,

Thanks for the attachments. Very nice literature on nonlocal models but these do not answer my question.

Prof. Paolo,

Thanks for sending the articles. Went through your articles briefly. This is more or less what I am tying to find. Regarding your statement

"You imagine, e.g. on the basis of tomography analyses or even just on direct observation, an initial distribution (the one that you use in deriving from a lattice scheme constitutive equations) and find the one at the instant t furnished by your field equations. Then you compare the predicted result with what emerges from standard traction tests, and you rearrange the initial ansatz about the defect distribution."

You say that we rearrange the initial ansatz while comparing the predicted distribution with that obtained from a standard traction test. How can we do this? Wouldn't  the crack density distribution be different under other/complex loading conditions. See "damage tensors and the crack density distribution. V. A. Lubarda and D. Krajcinovic, Int J. Solids Structures Vol. 30 No. 20 p.p. 2859-2877, 1993".

Your work is mainly related to (elasto-damage) brittle failure dominated by micro-cracks. As we know that the damage mechanism and evolution in brittle and ductile materials are quite different, how can we extend the order-parameter concept (multi-field theory) for ductile materials where we have elasto-plastic damage. Note that in ductile materials we (may) also have the phenomenon of plastic instability.

You are asking an inverse problem. Your traction-test data may just give you some indications, or at least they suggest the need to revise your initial ansatz, nothing more. The crack density distribution depends on the material's history. It's very hard to imagine determining exactly it from a single traction experiment. Just visual techniques, such as tomography (it seems to me that J. van Mier uses them experimentally) may give you an indication but only in statistica sense. Then you have to approximate the pictures you receive from the experiment and for them a tensor representation in the stereographic terms suggested by Ken Ichi Kanatani can be useful (there is an extended description of them in Krajcinovic's book on damage mechanics).

As regards the extension of what I sent you to the elastic-plastic setting is to consider elastic-plastic the rods in the lattice, identifying then not only the constitutive structures but also the flow rules. It is possible to do it. I did not write nothing about, being involved in other research engagements and topics so far, but I imagine what one should do. 

Thanks for your clarification. It is clear now. However, there must be numerical proof of the method that it works acceptably good for real life loading problems in terms of prediction. Acceptably good is indeed subjective but what I mean is not exact but close to reality.

Has the multi field theory been implemented in any commercial code yet or not? I can imagine that implementing this model via a user subroutine can be very challenging (or impossible)?

Thanks Prof. Paolo for the fruitful discussion. I specially enjoyed the philosophy of fitting experiments to models instead of model to experiments. Actually you are right, somewhere back in mind we fit experiments to models but we don't realize it.