An analytic derivation of the bifurcation conditions for localization in hyperelastic tubes and sheets

TL;DR

This paper analytically derives the conditions for localized bulging and necking in hyperelastic tubes and sheets, clarifying previous numerical findings and providing new insights into bifurcation criteria.

Contribution

It offers the first analytical derivation of bifurcation conditions for localization phenomena in hyperelastic structures, clarifying differences between tube and sheet cases.

Findings

Analytic bifurcation conditions for hyperelastic tubes and sheets.

Equivalence of bifurcation condition to Jacobian determinant vanishing in tubes.

Alternative interpretation for bifurcation in sheets.

Abstract

We provide an analytic derivation of the bifurcation conditions for localized bulging in an inflated hyperelastic tube of arbitrary wall thickness and axisymmetric necking in a hyperelastic sheet under equibiaxial stretching. It has previously been shown numerically that the bifurcation condition for the former problem is equivalent to the vanishing of the Jacobian determinant of the internal pressure $P$ and resultant axial force $N$, with each of them viewed as a function of the azimuthal stretch on the inner surface and the axial stretch. This equivalence is established here analytically. For the latter problem for which it has recently been shown that the bifurcation condition is not given by a Jacobian determinant equal to zero, we explain why this is the case and provide an alternative interpretation.