Localized edge state equilibria control when a soda can buckles

Thin-walled cylindrical shells such as rocket walls (or soda cans) offer exceptional strength-to-weight ratios yet predicting at which load the structure becomes unstable and fails remains an unsolved problem. Shells buckle and collapse at loading conditions much below those predicted by linear stability theory. We thus propose a fully nonlinear approach and show that fully nonlinear equilibrium states located on the boundary of the unbuckled state's basin of attraction define critical perturbation amplitudes and guide the nonlinear initiation of catastrophic buckling. For a clamped thin cylindrical shell under axial compression a fully localized single dimple deformation is identified as the edge state—the attractor for the dynamics restricted to the stability boundary. Under variation of the axial load, the single dimple undergoes homoclinic snaking in the azimuthal direction, creating states with multiple dimples arranged around the central circumference. Once the circumference is completely filled with a ring of dimples, snaking in the axial direction leads to further growth of the dimple pattern. The bifurcation structure of the equilibria closely resembles that observed in the Swift-Hohenberg equation with quadratic-cubic nonlinearity.