From turbulence transition to the buckling of a soda can

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. This failure of linear theory is traditionally ascribed to extreme sensitivity to unavoidable shell imperfections, which modify the linear thresholds and lead to unpredictable stochastic variations of buckling loads for nominally identical shell structures.

We propose a complementary fully nonlinear dynamical systems approach inspired by successful recent descriptions of the transition to turbulence in shear flows. We show both experimentally and theoretically that unstable 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. By following unstable equilibria including so-called edge states as a function of loading conditions using both numerical continuation and non-destructive experimental probing techniques, we characterize the load-dependent variations of the basin of attraction of the unbuckled state. Specifically, we identify an experimentally accessible representation of the hyperdimensional basin of attraction in the form of a low-dimensional stability landscape which fully encodes the stability of a cylindrical shell. Together with a characteriziation of typical perturbations, the mapping of basin boundaries opens an avenue for accurately predicting when an individual shell structure buckles.