The authors’ previous investigation [J. Elasticity 14, 103-126 (1984)] into the stability of a solid circular cylinder, made of a particular homogeneous and isotropic nonlinearly elastic material, under compressive end forces is here continued. The stored-energy function of the particular material used is \(\sigma(F)F\cdot F/2+(\det F)^{-m}/m\) (with F denoting the transplacement gradient and \(m>0)\). Starting from a trivial homogeneous solution to the relevant nonlinear boundary-value problem, the authors consider the superimposed linear problem under the restriction on axi-symmetric types of instabilities. By use of former results, the question of instability reduces to proving that a certain determinant, which corresponds to the traction-free condition, is zero at some critical value of the loading.
It is shown that for each eigenmode \(n=1,2,3,..\). there exists a unique loading value \(\lambda\) (n) that makes the determinant zero. The following conclusions are arrived at: (i) For a given value \(\lambda\) at most two eigenmodes occur. (ii) For sufficiently thick cylinders \(n=1\) is the eigenmode that occurs first. (iii) For sufficiently thin cylinders \(n=1\) is not the eigenmode that occurs first. (iv) There exist cylinders whose first point of instability occurs at two distinct eigenmodes simultaneously.