Expansion of a cavity in a rubber block under stress: application of the asymptotic expansion method to the analysis of the stability and bifurcation conditions

Abstract

In order to analyse the stability and bifurcation phenomena occurring during expansion of a small void in a rubbery material, the behaviour of spherical shells submitted to a combined far-field pressure and uniaxial tension has been investigated, considering a general nonlinear isotropic elastic compressible behaviour of the material and without any restrictions on the shell thickness. A radial solution for the deformation gradient with a spherical symmetry has been exhibited, which is valid for any behaviour law and consists of a homogeneous deformation. The three-dimensional problem is then linearized around this trivial solution, and we show the existence of a pressure interval containing the zero value, in which the solution is reduced to the trivial solution, which is therefore infinitesimally stable. The condition for stability obtained is compared with Hadamard's condition; particularly, it is shown that both are identical when the material is supposed to have a St Venant-Kirchhoff behaviour law. When the applied pressure lies outside the stability interval, we determine the bifurcation points of the shell around the trivial solution, first when only a pressure is applied and secondly when there is an additional far-field tension, much smaller than the applied pressure. The form of the stress distribution on the boundary of the cavity suggests a possible bifurcation of the spherical solution towards a family of axisymmetric solutions. Within this hypothesis, we get a relation between the geometrical parameter of the shell (its radius and thickness), the mechanical properties of the material and the critical load. The analyses provide evidence of the non-uniqueness of the bifurcation behaviour, since we exhibit some peculiar bifurcation points associated with an infinity of branches of axisymmetric solutions.