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Deformation 'Gradient', Right/Left Cauchy Green Compatibility

Amit Acharya's picture

I post some (hand-written) notes on compatibility conditions for both small and finite strains that I have used for helping me in lecturing. These may be useful for our student friends on imechanica. I also post a paper on compatibility conditions for the Left Cauchy-Green field in three dimensions as well as the paper by Janet Blume on the same subject.

 As you will see, Left Cauchy Green compatibility is really hard and I believe quite interesting from the nonlinear PDE point of view. Basically, it comes down to dealing with non completely integrable systems. I would like to go back to this sometime to get more out of the general result, but being an engineer does not allow such luxuries :) - one has to be relevant, whatever that means! More seriously, I think there is more to the utility of the result than meets the eye but this will require having a deeper understanding of the result which I don't think I have at this point. Perhaps Cartan's Method of Equivalence (the book by Gardner I refer to) can help, but Gardner's geometric exposition is beyond my mathematical competence at this time.

It also means that I should thank John Bassani here (my post-doctoral supervisor) who created the conditions under which I could find the time to think about these things without being bothered about hustling for funding or things non-academic, and to my two fantastic advisors, Tarek Shawki and Don Carlson, for the training that allowed me to even consider such questions.

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Amit Acharya's picture

I guess I should add here and in the thread "Strain compatibility equations in nonlinear solid mechanics" initiated by Ramdas Chennamsetti that the assertion that two deformations having the same Right Cauchy Green field differ at most by a rigid deformation depends in an important way on requiring that the C field (in Shield's proof) be twice continuously differentiable. In Janet Blume's work, she shows that the same result holds if the two deformations are merely regular (i.e. they are C^1, with positive determinant of the deformation gradient), but it is important that the mappings involve 3 dimensional bodies in three dimensional Euclidean space. For mappings of lower dimensional objects, if the two configurations set up by the two deformations are only related by a C^1 mapping, then this result might not hold true.

For 2-d surfaces, this can be inferred from the famous Nash-Kuiper C^1 embedding theorem, an interesting and completely counter-intuitive implication of which is that a spherical shell can be inserted into a another shell of arbitrarily small radius without self-intersection and without changing lengths of any curves on the original surface! (keep folding away at arbitrarily small scales!)

 

ramdas chennamsetti's picture

Dear Sir,

Thank you!! I got your attachments.

With regards,

- Ramdas

Amit Acharya's picture

Dear Ramdas,

If you read the notes and have any questions, please feel free to ask. I'll try to answer as I find time.

 - Amit 

Amit Acharya's picture

In the context of deformations of 3-d bodies, in my first note I mentioned the result of Blume which shows that if two deformations of a reference configuration are regular (C^1, detF >0) and they have the same right Caucht-Green field, then they are related by a rigid deformation.

A corollary is that if a deformation has zero Lagrangean strain (right CG = Identity), then its rotation tensor field is spatially uniform.

A result of Y. Reshetnyak proves that this result is true if the deformation is merely continuous.

These facts shows the impossibility of having stress free microstructure with one zero energy well in elasticity theory. Interestingly, in elastoplasticity the elastic deformation 'Gradient' in the multiplicative decomposition is not required to be compatible (so it is not any gradient in general and hence the quotes), so kinematically there is no restriction for the elastic deformation to display stress free microstructure even when the elastic energy density has only one well. Whether the kinetic relations of a particular elastoplasticity model allow such microstructure to form is another matter.

 

Amit Acharya's picture

Another interesting recent result related to rigidity of maps of 3-d domains into 3-d Euclidean space (the dimension of domain and target can actually be greater than or equal to 2) due to G. Friesecke, S. Muller, and R. D. James, derived in conjunction with rigorous derivation of nonlinear plate theories, is the following:

For each square integrable map with square integrable gradient, it is possible to find an associated rotation tensor (the approximating rigid deformation gradient) such that the L_2 distance of the gradient of the map from this rotation is bounded by a constant multiple of the L_2 distance of the gradient from the group of all rotations.

Thus, a corollary is that if at each x the gradient of the map is close to a rotation (not necessarily identical at all x), then there exists a *single* rotation such that the gradient of the map is close to this one rotation at almost all points.

Amit Acharya's picture

I've added a recent talk I gave on compatibility of strain measures in nonlinear continuum mechanics to the initial blog for this thread.

 

(oxford_B_compatibility_talk.pdf)

Dear Prof Acharya

Thank you very much for your invaluable notes you posted here. This is fantastic. I was looking for incompatibility equation in finite deformation theory and I scratched my head for long time. by reading your notes, I realized the reason and how is diffuclt in finite deformation. I have  a question and I appreciate you if we can comminucate on them

1-So what is the meaning of curl(curl(F)) or curl(curl(L))  in finite deformation theory? can we compare it with compatibility equation in infinitesimal strain theory?

Amit Acharya's picture

Dear Ramin,

I am glad you find the notes useful.

As for your question: what object do you have in mind for the quantity L?

Assuming F is a candidate for a possible deformation gradient field, whose compatibility for being so you are interested in probing, testing curl F = 0 suffices.

If you take the 4th order tensor R in notes 'lecture 5', defined in terms of the C field, substitute  2E + I = C, linearize the Eqn R(E) = 0, I think you will get the compatibility equation in the infinitesimal theory.

Hope this helps.

You'll find some of the same information in the Wikipedia article

 http://en.wikipedia.org/wiki/Compatibility_%28mechanics%29

The article is based in large part on Amit's notes.

-- Biswajit

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