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

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.