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# How to deal with the constitutive integration in plasticity?

Plastic theory has been developed systematically. As a beginner, however, some questions have confused me for a long time.

As is well known, the constitutive integration must be deal with in the flow theory. Instead of the explicit algorithm such as M-iteration, the implicit one - stress update algorithm - has been attractive for its simplicity, the most famous of which is rp (return mapping) for J2 flow theory.

However, compared with M-iteration, it seems to be difficult to use the stress update algorithm for a complicate yield condition. Is it right?

In addition, is it necessary to deal with the constitutive integration for the deformation theory?

I hope for help. Thanks.

## Re: Integration in plasticity

First, avoid the "deformation theory of plasticity ". It is unphysical and as Allan Bower says, "(

it) should really be called `junk plasticity,’ because like junk food, is"attractive but not good for you.

I'm not familiar with the term M-iteration. However, any explicit stress update algorithm requires very small time steps (often with subcycling) to maintain stability and accuracy. My approach has been to use backward Euler for the stress update for both explicit and implicit time stepping schemes.

If the yield condition is complicated then one does have to be careful to make sure that the return mapping is done correctly, e.g., by a closest point projection (

in the energy norm) of the trial stress onto the yield surface. However, once the algebra is worked out, the implementation is relatively straightforward for most phenomenological models. The algebra can be tedious, especially for finite plasticity with hyperelasticity (even for the isotropic case). The best textbook on the topic continues to be Simo and Hughes' "Computational Inelasticity".-- Biswajit

## Deformation theory is pretty good for analysis and understanding

Biswajit,

I think you are being too harsh in classifying deformation theory as you have above. We have two responsibilities in developing a model of physical phenomena - first it has to be physically 'correct' and, just as important, one has to be able to analyze the theory to understand qualitative features of solutions. A correct theory is 'useless' unless the qualitative features of its solutions can be determined. It is here that deformation theory plays a huge role in plasticity (at least in my opinion).

Most people who know plasticity will agree that because of the path independence of its total work potential, deformation theory is unphysical - but just because a theory of plasticity is path-dependent does not make it produce physical predictions. In fact, in certain situations when loading conditions satisfy the hypotheses of deformation theory, it produces more realistic results than flow theory, as is well known in the case of forming of thin sheets and the understanding of necking in them. Ordinary work-hardening J2 flow theory would predict no necking (and hence limit to formability) in the hardening range whereas J2 deformation theory does.

See the classic papers by

1) Christofferson and Hutchinson - J2 corner theory which is a flow theory that reduces to J2 deformation theory under proportional loading. The corner flow theory allows predictions of bifurcations because it allows for vertices on the yield function.

2) Storen and Rice - Analysis of necking of sheets using deformation theory.

So, what is my point? The point is, deformation theory is not useless, but very useful in understanding behavior of solutions of even flow theories and in motivating the development of the latter to include physically realistic effects. This is much like linear elastic fracture mechanics, or the theory of nonlinear hyperbolic equations where you allow shocks - not strictly physically true, but it would be unwise to deny their conceptual depth.

Just my opinion.

- Amit

## Deformation theory of plasticity

I agree that the deformation theory is useful in certain circumstances. However, I feel that numerical solutions of complicated problems is not one of those circumstances. My peeve with the deformation theory is that it will inevitably be applied to situations where it is not applicable, particularly in the hands of beginners. And it can be nontrivial to figure out whether such a theory is applicable or not in certain situations.

-- Biswajit

## Defmn. theory, numerics, complicated problems

Biswajit,

Given the work involved in putting together a robust numerical implementation and the generality one expects out of something requiring that kind of work, I agree that one should probably worry only with flow theories as a deformation theory is not generally applicable. I agree with you that without full knowledge of what one is doing, it may be dangerous for beginners to work with deformation theory.

Somewhat of an important aside to the specific discussion of constitutive integration:

Generality of a framework and the ability to produce 'numerical solutions to complicated problems' using it can sometimes be uncorrelated. Anything related to bifurcations/approach to such is probably the most complicated problem you can ask a numerical implementation of any theory to predict. And as I have mentioned before, some flow theories will simply not make the right predictions even though they are more 'general' in the sense we are discussing it, so you may question the generality. You will also know from experience that whenever one needs to solve a really complicated problem numerically, to gain real understanding, one almost always needs to know a fair bit analytically as to what to expect and guide the numerics in the right way. It is here that I feel that even in the case of numerical implementations of plasticity models, it is good to know/study about deformation theory results even if one does not implement it numerically.

Of course, this is perhaps an aside to the main discussion of this thread, but I think it is an important aside to consider.

- Amit

## Re:Deformation theory and gradient plasticity

Amit,

One of the problems with tackling bifurcation problems via ingenious models is that the physical cause of such bifurcations is not addressed adequately. Given that time is always in short supply, I would personally not spend too much time on deformation theories of plasticity even though they can be useful in gaining insights.

Coming back to the main point of this thread, I have spent a considerable amount of time trying to figure out which nonlocal/gradient theory of plasticity is the "best". My answers at this point are "hard to say" and "not obvious at all." The main reason is that few such theories are based on sound and consistent physical principles. I feel that we have, as a community, explored the top down approach quite a bit. At this point, it's time to explore bottom up approaches more thoroughly so that a few clear winners can emerge in the battle for nonlocal theories of plasticity.

-- Biswajit

## Re:Deformation theory and gradient plasticity

Biswajit,

"One of the problems with tackling bifurcation problems via ingenious models is that the physical cause of such bifurcations is not addressed adequately."

I wasn't talking about ingenious models. I was talking about the only models available. In the concrete context of understanding necking in thin sheets, what model would one apply? And remember J2 corner theory is motivated by a lot of very serious homogenization work on polycrystalline aggregates (Hill, Hutchinson) that shows that vertices on yield surfaces are a natural outcome of plastic deformation.

I thought the main point of this thread was simply ocnstitutive integration of plasticity and SG appeared later. Your statement about bottom-up approaches is laudable - I suppose we are all waiting for the truly bottom-up approach for homogenized response in plasticity that honestly faces up to the time-scale issue - if you start with MD it is at least 15 orders of magnitude; if it is DD then at least 9 orders of magnitude. Till then, (un)fortunately, models will have to do.

I think I have said all that I can say that is useful in this discussion, and I will retire here.

- Amit

## Re:Deformation theory and gradient plasticity

Amit,

I do appreciate your point. I am uncomfortable with theories that depend on sharp edges on the yield surfaces to deal with bifurcation but that may just be due to my incomplete understanding. You are right, we do the same thing for shocks and linear elastic fracture. I just feel that misleading conclusions can be drawn in some instances if one is not very careful.

When I wrote my first comment on this thread I was thinking of pressure dependent yield surfaces with caps for geomaterials as examples of complex yield conditions. In one of his later comments the author indicated that he was talking about gradient plasticity rather than complicated local plasticity models. Clearly several other issues arise in the constitutive integration of nonlocal models.

I'm not making much progress here either and will yield the stage for experts on that subject.

-- Biswajit

## About Top-down and Bottom-up approaches

First of all, Thank you and Acharya, your discussions are very useful for me.

I'm sorry that I didn't point out which yield condition I was concerned, however, the complicated models like geomaterials have also confused me for a long time. Thank you to clarify it.

In your latest comment, Top-down and Bottom-up approaches were proposed. Can I understand in this aspect: the approaches which have physical meaning are Bottom-up, such as MD, DD, and the phenomenological ones are Top-down?

## About Top-down and Bottom-up approaches

By "top-down" I mean approaches such as that in "Strongly non-local gradient-enhanced finite strain elastoplasticity" by Geers, Ubachs, and Engelen, (2003), IJNME, 56:2039-2068. In that particular paper the yield function is assumed to have the form:

The parameter is determined by phenomenological rules. For instance,

The parameter \kappa in turn is determined via a nonlocal field variable where this nonlocal field is the solution of a Helmholtz type equation

where is the local counterpart of .

Another formulation that depends directly on gradients is "Gradient-dependent plasticity:Formulation and algorithmic aspects" by de Borst and Muhlhaus, (1992), IJNME, 25:521-539. In that case, the Laplacian of the internal variable is used directly in the yield condition.

The question I have been unable to solve is how do you determine the form of the nonlocal extension and how do you the length scales, etc. at which these formulations are valid. These are examples of the top down approach.

To answer my questions one has to start from more fundamental premises. That involves, depending on how "fundamental" you want to go, on a large length of length scales. And, as Amit pointed out, just considering length scales is not enough. Dynamic processes are at work and you have to consider a huge range of time scales too.

Discrete dislocation plasticity or Amit's field theory of dislocations can be starting points for the bottom up approach. An early example of researchers trying to meet somewhere in the middle is "A comparison of nonlocal continuum and discrete dislocation plasticity predictions" by Bittencourt, Needleman, Gurtin, and van der Giessen, (2003), JMPS, 51:281-310. Since 2003, quite a bit of work has been done on the problem but everyone seems to be working on their own favorite idea and no consensus has yet emerged on the easiest/best way to tackle the problem.

I would like some of the experts on the subject (and I can't claim to be one) to elaborate on the state of the art and what they see is the way forward.

-- Biswajit

(P.S. Now that mimetex is no longer served by www.forkosh.com I think it's time that we had our own copy served by iMechanica. The process is simple. Just install mimetex on the iMechanica server under the cgi-bin directory. It takes a couple of minutes to install.)

Update:Following Yixiang's advice, I have started using wordpress to render my LaTex equations. The equations look nicer than those rendered by mimetex.

## constitutive integration of deformation theory

Acharya,My research topic is the strain gradient theory (SG). There are so many SG theories now, and I want to compare them with each other through FEM. For MSG theory (one of SG propose by H.Gao(1999)), it seems that the deformation theory is much easier to implement compared with the flow theory.I know that you have proposed some useful opinions of SG. Can you give me some suggestions? By the way, is it necessary to deal with constitutive integration for the deformation theory?Thanks.

## SG plasticity integration and sundry issues

For 'higher-order' strain gradient plasticity theories, you should check with Professor Young Huang (UIUC) for FEM implementations of both flow and deformation theories. You should also check with Profs. Garth Wells (Cambridge) and Krishna Garikipati (Michigan).

For the 'lower-order' gradient theory that Prof. John Bassani and I proposed, you simply cannot have a deformation theory. For the flow theory, if you go with an explicit update of the gradient variable, then there is almost nothing to do over the conventional theory. If you are interested, look at the papers by Acharya, Cherukuri and Govindarajan and the one by Tang, Acharya and Saigal. It is now clear that boundary conditions can be implemented for this model even in the 3-d case as has been shown the paper by Roy, Puri and Acharya. We now know a simpler, more efficient way to do the latter which we will write up some day.

One does not have to deal with constitutive integration in deformation theory. However, because it is nonlinear, implementing it is not very different from what one has to do in one increment of a flow theory with a backward Euler update, except the stress update in a flow theory is much more convoluted.

I hope this helps.

## LaTex

Biswajit:

Thank you for sharing the idea using LaTex. I'll try to learn it.

## How to deal with the constitutive integration in plasticity

I would like to suggest to see the following paper:

Y.Yamada ,Huang and I.Nishiguchi, Deformation theory of plasticity and its installation in the finite element analysis routine, in Numerical Methods in Fracture Mechanics(Pineridge, Swansea, 1980) pp. 343-357.

## Here is another thread of discussion on plasticity

Here is another thread of discussion on plasticity.