Stability and bifurcation of inflation of elastic cylinders

A method of obtaining a full (two-dimensional) nonlinear stability analysis of inhomogeneous deformations of arbitrary incompressible hyperelastic materials is presented. The analysis that we develop replaces the second variation condition expressed as an integral involving two arbitrary perturbations, with an equivalent (third-order) system of ordinary differential equations. The positive-definiteness condition is thereby reduced to the simple numerical evaluation of zeros of a well-behaved function. The general theory is illustrated by applying it to the problem of the inflation of axially stretched thick-walled tubes. The bifurcation theory of such deformations is well known and we compare the bifurcation results with the new stability analysis.