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# Deformation gradients and atomistic simulations

I came upon a recent paper called "Deformation gradients for continuum mechanical analysis of atomistic simulations" by Jonathan A. Zimmerman , Douglas J. Bammann, and Huajian Gao, International Journal of Solids and Structures 46 (2009) 238–253 where the authors conclude with

"Although the concept of the multiplicative decomposition of the deformation gradient within the context of plasticity theory was discussed and considered, our simulation results show zero values of curl within material containing defects such as dislocation loops and stacking faults. As such, it appears that our atomic-scale metric captures the full, compatible deformation field that the material is subject to. As noted above, the deformation gradient expression itself produces discontinuities reflective of material that contain defects such as partial dislocations and stacking faults. These discontinuities would also exist for regions through which a full dislocation has traveled and no stacking fault or other defect remains (Zimmerman, 2000). In both instances, it may be possible to use geometric information about the material defects created (such as the Burgers vectors as quantified by the slip vector) to isolate the ‘plastic’ component of the deformation gradient Fp, thereby enabling Fe to also be determined. However, it is not yet apparent how to perform this decomposition and further work in this area is also warranted."

Can someone parse the significance of this for me? Amit? Anyone?

-- Biswajit

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## Comments

## Hartley Mishin

Biswajit,

I was not aware of this paper - thanks for pointing it out.

I guess the authors of the papers are in the best position to parse the comment.

As far as I am concerned, that the total deformation gradient (with the authors' definition) turns out to be compatible in most circumstances is all consistent and no surprise. What is also important is to be able to identify the inverse elastic distortion and then be able to calculate its curl on the only physical reference available - the current. That this can be done more or less unambiguously when you have a basic unstressed crystal structure in mind can be seen from the paper (apparently the current authors are not aware of the following work):

Hartley, C. S., Mishin, Y. 2005 "Characterization and visualization of the lattice misfit associated with dislocation cores, Acta Materialia, 53, 1313-1321.

This paper shows that concepts like the Nye tensor are spot on in the case of defects that they characterize from MD (I believe) simulation results.

Unfortunately, the paper has some typos which make the mathematical exposition confusing, but one can sort through that with a bit of patience.

Finally, I would mention that it is important to keep distinct the meaning of slip and dislocation, the latter being the boundary between differently slipped regions.

Lots to get into, not enough time....

Sorry to be of not much help.

- Amit

## discrete compatibility

Dear Biswajit:

Thanks for pointing out this paper and also thanks to Amit for his thoughtful comments.

Of course, I can't comment on the significance of this work but have a couple of quick comments.

1) Compatibility is discussed in terms of an equivalent continuum. As far as I understand this, this is vague and may depend on the way the "atomic-scale deformation gradient" is defined. I think one should start with discrete problem ab initio. The following work is relevant here:

Ariza MP and Ortiz M, Discrete Crystal Elasticity and Discrete Dislocations in Crystals, Archive for Rational Mechanics and Analysis 178: 149-226, 2005

1) In 1-D, compatibility is trivial. I don't understand why the atomic chain example is mentioned in 5.1.

Regards,

Arash

## Fe-1 in continuum plasticity

Biswajit,

Perhaps I can add a relevant comment. On page 244, the authors say

" F^e refers to the reversible deformations caused by the loading of the material" and then, consistent with this definition, on page 250 they take off loads on a deformed specimen (presumably containing resdual stress inducing defects) and call the deformation gradient between the loaded and unloaded configurations F^e-1.

Since by definition the loaded and the unloaded configurations are coherent global configurations, the curl of the defmn. gradient field between these configurations better come out to be zero, and it does. Also, by definition, this always has to be the case (regardless of loading, material, and everyhting else) even in conventional continuum plasticity employing the multiplicative decomposition. Note that were this the definition of F^e-1 and with the multiplicative decomposition being valid, F^p would *always* be compatible in conventional continuum plasticity.

The point is - in continuum plasticity, the field defined in the paper as Fe^-1 is not what is meant as the inverse elastic distortion. The unloaded configuration is generally stressed due to the presence of defects.

What corresponds to F^e-1, say at a given point, in continuum plasticity is a composition of the deformation gradient at that point between the loaded and unloaded configurations composed with another invertible tensor that represents locally completely unstressing an infinitesimal neighborhood around the point if you wish (and realigning it etc., etc.), as in the eqn. 53 of the paper - only, in continuum plasticity these two are clubbed together and this is actually adequate even in situations when stress fields of dislocations have to be calculated.

So some differences in terminology, I guess.

- Amit

## Re: Fe^-1 and compatibility

Amit and Arash,

Thanks for your comments and pointers. Back to the library for now.

-- Biswajit

## Comments on the atomic-scale deformation gradient

Hi everyone,

Thanks for the interesting and lively discussion of the paper. I'm glad to see it evoking much thought, commentary and deliberation, the objective of every technical article. While I don't have time for a complete discussion here, I will be glad to address a few of the points raised.

Regarding the signficance of the paper, it was our (my coauthors and myself) objective to present/revisit a definition for a deformation measure from atomistic quantities and to examine this definition from the perspective of the mechanician. While many have defined similar measures, not nearly enough have looked at how rigorous their measure compares with its continuum counterpart. Since a key property of a deformation gradient is compatiblity, where possessing or lacking such compatibility each have their own interpretations, it seemed reasonable to examine this with our atomic-scale metric.

Regarding the work by Hartley and Mishin, we were not aware of it while writing the article, but have since become aware of it especially after having a few, brief conversations with Craig Hartley himself. Reference to this work, and his previous articles, will be made in future publications.

Finally, regarding the comment about our 1D example, I would submit that what is trivial to one reader is not true for all. Further, it is often the case that seeming "trivialities" are often left out of a manuscript only to be brought up during the review process. It was our attempt to be comprehensive rather than not.

Thanks again for the interest,

Jon Zimmerman

## Compatibility in 1D

Dear Jon:

Thank you for your detailed response to all the comments.

By "trivial" I didn't mean the problems discussed are trivial. In dimension one, any given strain field can be integrated to find a single-valued displacement field. This is why in elasticity compatibility equations are relevant only in dimensions two and three.

In a Riemannian manifold, curvature tensor is the key geometric object for local flatness. Vanishing of this 4-tensor guarantees local flatness. In dimension three, it turns out that vanishing of a "simpler" quantity called Ricci curvature would suffice (a symmetric tensor and hence six compatibility equations). In dimension two having Gauss curvature would suffice (so only one compatibility equation). Any one-dimensional manifold is locally flat. So, in this sense, there is no need to check compatibility in a one-dimensional problem.

For 1D problems a decomposition of deformation gradient into elastic and inelastic (plastic, growth, etc) parts is equivalent to a similar decomposition of deformation mapping as a direct consequence of this fact. The following paper implicitly uses the local flatness of one-dimensional manifolds.

Senan, N. A. F. and O’Reilly, O. M. and Tresierras, T. N. [2008], Modeling the growth and branching of plants:

A simple rod-based model. Journal of the Mechanics and Physics of Solids 56:3021-3036.

Having said all this, I agree with you that explaining all the details and sometimes even the things that may seem obvious or classic to some people is a wonderful thing to do as it can help readers.

Regards,

Arash

## coarse grained model based on atomic deformation measure

Hi all,

To follow up on comments to Jon’s IJSS paper and show another aspect of the deformation measure proposed in his paper, we have developed a coarse-grained model for pi-electron based carbon nanostructures. The proposed deformation measure, which we termed “spatial secant”, is the same in spirit as the first order expansion in Jon’s paper. We established a discrete version of the hyperelastic model based on this measure and have shown that the model is quite robust in describing the mechanics of CNTs through numerical simulations. More details on the computational implementation and validations can be found in the following reference.

Qian, D., Q. Zheng and R. S. Ruoff "multiscale simulation of nanostructures based on a discrete hyper-elastic approach"

Computational Mechanics, 42(4): p. 557-567, 2008Comments on the paper are of course welcome.

Dong Qian