Axisymmetric Instabilities for Elastic Conical Shells under Compressive End Loadings

Abstract

We deal with the problem of equilibrium and buckling of nonlinear elastic axisymmetric shells, whose referential shape is a truncated circular cone, subject to compressive end loadings. It is a non-trivial generalization of the cylindrical case, already studied by Cohen and Pastrone; in this particular case, suitable constitutive restrictions, which are not needed for cylinders, must be satisfied to allow the conical shell to sustain end loadings, even small, without bending. In particular, we consider thin Kirchhoff shells and prove, by means of the bifurcation theory of Poincaré, the non-uniqueness of solutions of the boundary value problem associated with the equilibrium equations, the assigned end loadings and the geometrical constraints: i.e., the axisymmetry and the inextensibility along meridians. The critical loads are determined as well as the bifurcation points. If the material is hyperelastic the equations of equilibrium are derived from a variational principle and, for some special form of the strain energy function, a Hamiltonian formulation can be provided. The possibility of a non-convex strain energy function is briefly discussed.