On a consistent finite-strain shell theory for incompressible hyperelastic materials

Abstract

In this paper, a consistent finite-strain shell theory for incompressible hyperelastic materials is formulated. First, for a shell structure made of an incompressible material, the three-dimensional (3D) governing system is derived through the variational approach, which is composed of the mechanical field equation and the constraint equation. Then, series expansions of the independent variables are conducted about the bottom surface and along the thickness direction of the shell. The recursive relations of the coefficient functions in the series expansions can be derived from the original 3D governing system. Further from the top surface boundary condition, a 2D vector shell equation is obtained, which represents the local force-balance of a shell element. The associated edge boundary conditions are also proposed. It is verified that shell equation system is consistent with the 3D variational formulation. The weak formulation of the shell equation is established for future numerical calculations. To show the validity of the shell theory, the axisymmetric deformations of a spherical and a circular cylindrical shell made of incompressible neo-Hookean materials are studied. By comparing with the exact solutions, it is shown that the asymptotic solutions obtained from the shell theory attain the accuracy of O(h2).