Structural localization phenomena and the dynamical phase-space analogy

Abstract
The localization of buckle patterns in elastic structures is reviewed from three complementary viewpoints: (a) from a modal perspective, (b) from a formulation which allows the amplitude to modulate in an asymptotically defined ‘slow’ space and (c) from a dynamical analogy in phase space suggested by the form of the underlying differential equation. A simple strut on an (asymmetric) nonlinear foundation provides a typical illustrative example. The three approaches emphasize different features of the localization phenomenon. The modal view illustrates the distinctive effects of boundary conditions, the modulated approach generates a convenient second-order differential equation in the amplitude function and the dynamical phase-space analogy suggests a useful interpretation of localization as a homoclinic connection. Comparisons are also made with nonlinear numerical solutions. As the strut length approaches infinity it is shown that the fully localized solution represents the unstable post-buckled state with the lowest energy, allowing evaluation of the minimum energy barrier relevant to dynamical impact studies. Attention is drawn to the possibility of spatial chaos, becoming manifest as a randomly spaced sequence of localizations caused by a regular sinusoidal spatial imperfection.

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