Asymptotic Analysis of the Eversion of Nonlinearly Elastic Shells II. Incompressible Shells

This paper treats the eversion of axisymmetric, strictly convex, incompressible nonlinearly elastic shells within a general geometrically exact theory in which the shell can suffer flexure, shear, and both midsurface and transverse extension. The governing equations differ considerably from those for compressible shells. We first formulate the governing equations carefully, showing how to handle the 3-dimensional notion of incompressibility, and paying special attention to the constitutive equations. We prove that when a thickness parameter δ is sufficiently small, there is an everted state, having a lip near the edge, that can be approximated effectively by an asymptotic series whose error we estimate.