Rigorous asymptotic analysis of buckling of thin-walled cylinders under axial compression

TL;DR

This paper provides a rigorous asymptotic analysis of buckling in thin-walled cylinders under axial compression, establishing the critical load, bifurcation modes, and stability criteria using advanced mathematical tools.

Contribution

It introduces a rigorous constitutive linearization approach, sharp estimates of Korn constants, and develops the buckling equivalence concept for cylindrical shells.

Findings

Critical load coincides with bifurcation load in shell theory

Linear bifurcation modes minimize the second variation asymptotically

Asymptotic estimates of Korn constants are established

Abstract

Using rigorous constitutive linearization of second variation introduced in [6] we study weak stability of homogeneous deformation of the axially compressed circular cylindrical shell, regarded as a 3-dimensional hyperelastic body. We show that such deformation becomes weakly unstable at the citical load that coincides with value of the bifurcation load in von-K\'arm\'an-Donnel shell theory. We also show that the linear bifurcation modes described by the Koiter circle [11] minimize the second variation asymptotically. The key ingredients of our analysis are the asymptoticaly sharp estimates of the Korn constant for cylindrical shells and Korn-like inequalities on components of the deformation gradient tensor in cylindrical coordinates. The notion of buckling equivalence introduced in [6] is developed further and becomes central in this work. A link between features of this theory and sensitivity of the critical load to imprefections of load and shape is conjectured.