Asymptotic results for bifurcations in pure bending of rubber blocks

TL;DR

This paper analyzes the bifurcation behavior of incompressible neo-Hookean rubber blocks under pure bending, revealing the transition from Euler buckling to surface instability through asymptotic and numerical methods.

Contribution

It provides a detailed asymptotic analysis of bifurcations in rubber blocks, connecting buckling and surface instability phenomena with novel analytical and numerical insights.

Findings

Identifies Euler-type buckling for finite thickness ratios

Shows transition to surface instability as thickness ratio increases

Validates analytical results with numerical simulations

Abstract

The bifurcation of an incompressible neo-Hookean thick block with a ratio of thickness to length {eta}, subject to pure bending, is considered. The two incremental equilibrium equations corresponding to a nonlinear pre-buckling state of strain are reduced to a fourth-order linear eigenproblem that displays a multiple turning point. It is found that for 0 < {eta} < {infty}, the block experiences an Euler-type buckling instability which in the limit {eta} -> {infty} degenerates into a surface instability. Singular perturbation methods enable us to capture this transition, while direct numerical simulations corroborate the analytical results.