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This and the next 5 images are frrom:
Tobias Kreilos and Tobais M. Schneider, “Fully localized post-buckling states of cylindrical shells under axial compression”, Proceedings of the Royal Society A: Mathematial, Physical and Engineering Sciences, Vol. 473, No. 2205, September 2017, DOI: 10.1098/rspa.2017.0177
ABSTRACT: We compute nonlinear force equilibrium solutions for a clamped thin cylindrical shell under axial compression. The equilibrium solutions are dynamically unstable and located on the stability boundary of the unbuckled state. 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. These fully nonlinear solutions embedded in the stability boundary of the unbuckled state constitute.critical shape deformations. The solutions may thus be a step towards explaining when the buckling and subsequent collapse of an axially loaded cylinder shell is triggered.
Definition of “edge state”: If the compressive load is below the critical buckling load, small deformations will decay towards the unbuckled state. The set of all these deformations that return to the unbuckled state is its basin of attraction, a region in state space that surrounds the unbuckled state. States that lie outside this basin will not return to the unbuckled state and almost all of them will evolve towards collapse. There is a boundary between these states and the basin of attraction of the unbuckled state—states in this boundary are the objective of this study as they may guide the transition to collapse. In the energy landscape, the linear stability of the unbuckled state is located at the bottom of a valley. The basin boundary corresponds to a mountain ridge surrounding it. States ON the boundary will slide down neither on one side of the ridge towards the unbuckled state nor on the other side towards collapse. They may, however, still evolve ON the basin boundary until they reach a saddle point. Such a saddle point is an equilibrium state and was termed mountain pass by Horák et al. In dynamical systems theory it is an edge state of the first-order system.
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