Finite elastic eversion of a thick-walled incompressible spherical cap

Abstract

For perfectly elastic rubber-like materials, which are capable of undergoing extremely large deformations, the number of exact solutions remains limited, especially in the context of fully three-dimensional deformations. Here a simple exact solution describing the finite elastic eversion of a sector of a thick-walled incompressible spherical shell is determined for the modified Varga elastic material. This new solution, which describes a portion of a spherical shell being turned inside out, is deduced from a known simplified system and it is shown, by solving the full equilibrium equations, that no further solutions of this type can be deduced for this particular material. Further, a general family of response functions is considered, which involves an arbitrary index n, and which incorporates standard materials such as the neo-Hookean and Varga strain-energy functions. It is established that other than n=1 (namely the Varga material) only the special case n=2 admits nontrivial solutions to the eversion problem, but the resulting second-order highly nonlinear ordinary differential equation appears not to admit any simple analytical solutions. Finally, the new solution is examined as a potential solution of the 'snap-buckling' problem of a spherical cap. Unfortunately, the solution appears not to be applicable to this problem and instead it is presented in the specific context of the eversion of a thick-walled spherical cap, with no applied forces acting on one of the surfaces of the deformed configuration.