An Analysis of the Linearized Equations for Axisymmetric Deformations of Hyperelastic Cylinders

Abstract

The equations describing the axisymmetric deformations of a cylindrical body composed of a hyperelastic isotropic material define a system in two variables (radius and height due to axisymmetry) of quasilinear partial differential equations subject to nonlinear mixed boundary conditions. The author considers the boundary value problem of specifying the displacement of the lateral surface of the cylinder subject to zero normal stresses on the top and bottom. It is shown that this problem admits a trivial solution consisting of a uniform expansion or compression in the radial and height directions. The author studies the linearization of the full nonlinear equations about the trivial solution and constructs solutions for the resulting system of linear partial differential equations. As a consequence of these explicit representations, one gets the characteristic equation defining the eigenvalues of the linearized problem, which represent bifurcation points of the nonlinear system. The corresponding eigenfunctions can be classified into those that are symmetric about the z = 0 axis (midplane of the cylinder) representing either necked or barreled states of the cylinder and those that break this symmetry. For a class of Hadamard-Green type materials and all cylinder heights, the existence of eigenvalues for the symmetry-preserving and symmetry-breaking characteristic equations is shown.