On non-linear stability in unconstrained non-linear elasticity

A method of obtaining a full three-dimensional non-linear Hadamard stability analysis of inhomogeneous deformations of arbitrary, unconstrained, hyperelastic materials is presented. The analysis is an extension of that given by Chen and Haughton (Proc. Roy. Soc. London A 459 (2003) 137) for two-dimensional incompressible problems. The process that we present replaces the second variation condition expressed as an integral involving a quadratic in three arbitrary perturbations, with an equivalent sixth-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 a thick-walled spherical shell. The present analysis provides a simpler alternative approach to bifurcation problems approached by using the incremental equations of non-linear elasticity.