Buckling of thin-walled cylinders from three dimensional nonlinear elasticity

TThe present work addresses the rigorous derivation of the Flügge treatment of the buckling of a thin cylinder. The incremental equilibrium equations in terms of generalized stresses are rigorously derived in terms of mean quantities (holding true regardless of the thickness of the cylinder), through a generalization of the approach introduced by Biot (1965) for rectangular plates. The incremental kinematics is postulated through a novel deduction from the deformation of a two-dimensional surface, thus generalizing an approach introduced to derive the incremental kinematics of a plate. The nonlinear elastic constitutive equations proposed by Pence and Gou (2015), describing a nearly incompressible neo-Hookean material, are used in a rigorous way. While the employed kinematics coincides with that used by Flügge, the incremental equilibrium and constitutive equations derived in this work are different from those given by Flügge, but are shown to reduce to the latter by invoking the smallness of the cylinder wall. The equations derived for the incremental deformation of prestressed thin cylindrical shells are general and can be used for different purposes. The study of the bifurcation problem of a thin-walled circular cylinder subject to compressive load is offered. When compared, the bifurcation landscape obtained from the formulation developed in this work and that given by Flügge are numerically shown to coincide and be consistent with results obtained by a fully three-dimensional theory of nonlinear elasticity. Furthermore the formula for the axial buckling stress of a ‘mid-long’ cylindrical shell made of a nearly incompressible neo-Hookean material and of a Mooney-Rivlin material are rigorously obtained from the presented formulation.