Deformation 'Gradient', Right/Left Cauchy Green Compatibility

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!)

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 

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.

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.

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 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.