On Constructing the Analytical Solutions for Localizations in a Slender Cylinder Composed of an Incompressible Hyperelastic Material

TL;DR

This paper develops analytical solutions for localization in a slender incompressible hyperelastic cylinder under tension, revealing how localization width depends on material parameters and predicting failure conditions.

Contribution

It introduces a novel coupled series-asymptotic expansion method to derive explicit analytical solutions for localization phenomena in hyperelastic cylinders.

Findings

Localization zone width depends on material parameters

Snap-back phenomenon occurs for small radius-length ratios

Analytical failure criterion predicts onset of material failure

Abstract

In this paper, we study the localization phenomena in a slender cylinder composed of an incompressible hyperelastic material subjected to axial tension. We aim to construct the analytical solutions based on a three-dimensional setting and use the analytical results to describe the key features observed in the experiments by others. Using a novel approach of coupled series-asymptotic expansions, we derive the normal form equation of the original governing nonlinear partial differential equations. By writing the normal form equation into a first-order dynamical system and with the help of the phase plane, we manage to solve two boundary-value problems analytically. The explicit solution expressions (in terms of integrals) are obtained. By analyzing the solutions, we find that the width of the localization zone depends on the material parameters but remains almost unchanged for the same material in the post-peak region. Also, it is found that when the radius-length ratio is relatively small there is a snap-back phenomenon. These results are well in agreement with the experimental observations. Through an energy analysis, we also deduce the preferred configuration and give a prediction when a snap-through can happen. Finally, based on the maximum-energy-distortion theory, an analytical criterion for the onset of material failure is provided.