Extension, inflation and circumferential shearing of an annular cylinder for a class of compressible elastic bodies

Abstract

This paper studies the classical problem of extension, inflation, and circumferential shearing of an annular cylinder for a new class of compressible elastic bodies wherein the left Cauchy–Green deformation tensor is given as a function of the Cauchy stress tensor. We use a semi-inverse method to study the problem by assuming forms for both the deformation field and the stress field. Focusing our attention on to three specific constitutive relations and two geometries corresponding to thick and thin annular cylinders, we study the qualitative features of the governing differential equations. The models are chosen so that they exhibit qualitatively different response features, one of them displaying a limiting stretch. The classical assumption that the hoop and axial stresses are nearly constant through the thickness of the thin annular cylinder subjected to inflation holds for this class of elastic bodies too. However, for thick-walled annular cylinders subjected to inflation at constant length and for a class of models that exhibits limiting stretch we find that stress “boundary layers” form and that the radial stretch is not monotonic.