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Deformation 'Gradient', Right/Left Cauchy Green Compatibility
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.