Branching of the radial expansion history during inflation of a non-linear incompressible isotropic time-dependent thick-walled sphere

Abstract

The spherically symmetric expansion of an incompressible isotropic spherical shell from pressure on its inner surface is considered. For any such material, the pressure—radius relation is given by an integral over the shell that depends on its constitutive response in equal biaxial stretch and the stretch distribution in terms of the radius of the inner surface. In this work, a general non-linear single-integral constitutive relation describes the equal biaxial stretch response. This relation includes two models of non-linear viscoelasticity and one model for scission and crosslinking of elastomers as specific cases. It results in a non-linear Volterra integral equation for the radius of the inner surface when the pressure is specified. The possibility is considered that the solution to the integral equation develops another branch at some time. A condition is derived that determines if and when branching will occur. It is shown that when the material is elastic, the branching condition reduces to the condition derived by Carroll for a non-monotonic pressure—radius relation. Some results are provided for the case of time-dependent materials, when the branching condition depends on the response to equal biaxial stretch histories.