Buckling of thin-walled cylinders from three dimensional nonlinear elasticity

TL;DR

This paper extends Flügge's classical buckling analysis of thin-walled cylinders by incorporating nonlinear hyperelastic constitutive laws, enabling more accurate modeling of materials like biological tissues and soft robotics.

Contribution

The authors re-derive Flügge's formulation to include any nonlinear hyperelastic constitutive law, broadening its applicability beyond the original incremental approach.

Findings

Confirmed Flügge's bifurcation results under simplified assumptions

Extended the theory to nonlinear hyperelastic materials

Enabled efficient computational analysis for complex materials

Abstract

The famous bifurcation analysis performed by Fl\"ugge on compressed thin-walled cylinders is based on a series of simplifying assumptions, which allow to obtain the bifurcation landscape, together with explicit expressions for limit behaviours: surface instability, wrinkling, and Euler rod buckling. The most severe assumption introduced by Fl\"ugge is the use of an incremental constitutive equation, which does not follow from any nonlinear hyperelastic constitutive law. This is a strong limitation for the applicability of the theory, which becomes questionable when is utilized for a material characterized by a different constitutive equation, such as for instance a Mooney-Rivlin material. We re-derive the entire Fl\"ugge's formulation, thus obtaining a framework where any constitutive equation fits. The use of two different nonlinear hyperelastic constitutive equations, referred to compressible materials, leads to incremental equations, which reduce to those derived by Fl\"ugge under suitable simplifications. His results are confirmed, together with all the limit equations, now rigorously obtained, and his theory is extended. This extension of the theory of buckling of thin shells allows for computationally efficient determination of bifurcation landscapes for nonlinear constitutive laws, which may for instance be used to model biomechanics of arteries, or soft pneumatic robot arms.