Investigating imperfection-sensitivity in shell buckling using a toy model

Shell buckling is known for its extreme sensitivity to initial imperfections. It is generally understood that this sensitivity is caused by an unstable (subcritical) bifurcation, i.e. geometric imperfections rapidly erode the buckling load of the perfect shell. It is less commonly appreciated that subcriticality also creates a strong proclivity for spatially localised buckling modes. The ability of localisations to appear anywhere across the domain (spatial multiplicity) leads to a large set of possible trajectories to instability, with each trajectory affine to a particular imperfection signature. Using a toy model of a link system on a softening elastic foundation, we show that the spatial multiplicity of localisations leads to a large spread in buckling loads, even for indistinguishable random imperfections of the same amplitude. By imposing a dominant imperfection, the strong sensitivity to random imperfections is ameliorated, and the ability to control the trajectory to buckling via dominant imperfections or elastic tailoring, creates interesting possibilities for designing imperfection-insensitive shells.