Real Shells Exhibit a Universal Localized Buckling Mode with Marginal Imperfection Dependence, Part I: Theory and Simulations

Imperfections are known to play an essential role in the buckling of a thin shell, but how they interact to control the onset of failure remains unclear.  We argue that the von Kármán-Donnell equations, accounting for the shells’ underlying geometric defect structure (w₀), are sufficient to predict the dynamics through buckling by comparing their numerical solutions to geometrically non-linear finite-element simulations. Both reveal that shells modeled with geometric imperfections predominantly exhibit localized buckling eigenmodes. Prior to failure, the amplitude of equilibrium radial deformations in the shape of the localized buckling eigenmode is well modeled by a saddle-node bifurcation.

Using only the experimentally determined geometric defect structure, we accurately predict the location of the localized buckling mode observed in high-speed videography. That is to say, an incomplete description of an imperfect shell’s underlying defect structure still provides meaningful predictions for its failure properties. These results point to a paradigm shift in our understanding of the role of localization and imperfections in the failure of thin shells which could have significant practical applications for the design of thin-walled structures.