Abstract
Shell buckling is known for its extreme sensitivity to initial imperfections. It is generally understood that this sensitivity is caused by subcritical (unstable) buckling, whereby initial geometric imperfections rapidly erode the idealized buckling load of the perfect shell. However, it is less appreciated that subcriticality also creates a strong proclivity for spatially localized buckling modes. The spatial multiplicity of localizations implies a large set of possible trajectories to instability-also known as spatial chaos-with each trajectory affine to a particular imperfection. Using a toy model, namely a link system on a softening elastic foundation, we show that spatial chaos leads to a large spread in buckling loads even for seemingly indistinguishable random imperfections of equal amplitude. By imposing a dominant imperfection, the strong sensitivity to random imperfections is ameliorated. The ability to control the equilibrium trajectory to buckling via dominant imperfections or elastic tailoring creates interesting possibilities for designing imperfection-insensitive shells.
Highlights
It is well known that systems governed by subcritical bifurcations are sensitive to initial imperfections
External perturbations or geometric imperfections round off the perfect bifurcation into limit points with rapidly decreasing magnitude of the critical parameter
We demonstrate how spatial chaos is a governing factor in the stochasticity observed in shell buckling, i.e., the large spread in buckling loads for seemingly indistinguishable imperfections of the same amplitude
Summary
It is well known that systems governed by subcritical bifurcations (unstable branching) are sensitive to initial imperfections. Geometric imperfections spanning the vector space of critical eigenvectors are conducive to premature buckling [4] Within this framework, stochastic variations in observed buckling loads are explained by different imperfection modes and varying imperfection amplitudes. Depending on the precise nature of initial imperfections and external perturbations, the cylinder can transition out of the prebuckling well via one of these many routes This situation of an infinite set of escape routes—with each trajectory to buckling sensitively dependent on the initial conditions—has certain analogies with temporal chaos and is often referred to as “spatial chaos” [9,10,11,12]. We demonstrate how spatial chaos is a governing factor in the stochasticity observed in shell buckling, i.e., the large spread in buckling loads for seemingly indistinguishable imperfections of the same amplitude
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