Snaking and laddering in axially compressed cylinders

Recent experimental and numerical results have illustrated the importance of spatially localised buckling modes in the loss of stability of axially compressed cylinders. Particular interest has focused on a single localised dimple that circumferentially multiplies along a post-critical equilibrium path of oscillating end-shortening—a phenomenon known as snaking. In contrast, multi-dimple buckling modes and their snaking equilibrium paths have not been studied in detail. In this work, two spatially localised modes—namely, a single dimple and two adjacent dimples—are identified as fundamental “building blocks” that can be repeated in intervals around the cylinder circumference to create different multi-dimple patterns and ensuing snaking sequences. Each snaking sequence circumferentially multiplies the starting dimple pattern until one ring of diamond-shaped buckles, ranging from eight to twelve waves, is complete. This set of five re-stabilised post-buckling modes matches the diamond-shaped rings of buckles observed by Yamaki in his seminal experiments, and uncovers their origins for the first time. By changing the applied loading from rigid to dead loading, however, the formation of a single dimple does not lead to a ring of buckles but to kinking of the cylinder. We show that snaking is possible when pre-buckling and periodic post-buckling states co-exist at the same total potential and localised buckles can be inserted between the two competing modes at little energetic cost. Overall, the findings strengthen the contemporary perspective that spatial localisation plays a key role in the buckling and post-buckling behaviour of axially compressed cylinders.