Suppose I have a transversely isotropic, hyperelastic material with known strain energy that is a fibrous composite. I am interested in an explicit formula for the displacements (so I can get the stress and strain tensors) for a pressure vessel made of such a material, but where the local fiber direction (i.e. axis of transverse isotropy) varies smoothly as a function of the radial direction ($f(r):\mathbb{R}\rightarrow\mathbb{R}^3$ is $C^\infty$).

For example, something like Roy, 1991, "Strength analysis and design of multilayered thick composite spherical pressure vessels" has (somewhat) explicit formulas. But note that my material does not change throughout the wall, except for the continuously varying local direction of anisotropy. Also, I do not have discrete layers and my material is non-linearly elastic.

This sounds like a problem for multilayer, laminar composites, but I do not have discrete layers, which those analyses that I have seen so far tend to assume (and reasonably so, since I am considering a model of a biological tissue, rather than something engineering oriented). I could also assume that there are a finite number of layers, but I don't think that would be analytically tractable, and my goal is to do some variational analysis over the space of functions $f$.

If it makes it easier, it would also suffice to have some analysis of a thick-walled "plate" with the properties above or even to assume linear elasticity.

Mainly, I am looking for literature (since, at least for me, this is not so trivial). But any help would be appreciated. Even better words to search for would be nice, since I am relatively new to continuum mechanics.


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