### Incompressibility in rod and shell theories

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We treat the problem of constructing exact theories of rods and shells for thin incompressible bodies. We employ a systematic method that consists in imposing constraints to reduce the number of degrees of freedom of each cross section to a finite number. We show that it is very difficult to produce theories that exactly preserve the incompressibility and we show that it is impossible to do so for naive theories. In particular, many exact theories have nonlocal effects.

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