On the role of localised post-buckling equilibria in axially compressed cylinders*

We revisit buckling of axially compressed cylinders by considering fully localised post-buckling states in the form of one or multiple dimples. Using a combination of nonlinear quasi-static finite element methods and numerical continuation algorithms, we trace the evolution of odd and even dimples into one ring of circumferential diamond waves. The growth of the post-buckling pattern with varying compression is driven by a homoclinic snaking sequence, with even and odd dimple solutions intertwined. The initially stable and axially localised ring of circumferential diamonds destabilises at a pitchfork bifurcation to produce a second circumferential snaking sequence that results in the Yoshimura pattern. Localised dimple solutions represent saddle points in the energy landscape providing an exponentially decreasing energy barrier between the stable pre-buckling and re-stabilised post-buckling wells. The significance of the Maxwell load as a measure for quantifying the onset of mountain-pass solutions and the reduced resilience of the pre-buckling state is assessed. Finally, conservative buckling loads for design are inferred by tracing critical boundaries of the snaking set.