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Effects of exponentially small terms in the perturbation approach to localized buckling

journal contribution
posted on 2023-05-18, 19:08 authored by Wadee, MK, Andrew BassomAndrew Bassom

Localized buckling solutions are known to exist in the heuristic model of an elastic strut resting on an elastic (Winkler) foundation. The primary localized solution emerging from the critical buckling state consists of an amplitude envelope of approximately the form of a hyperbolic secant function which modulates a fast-varying sinusoidal oscillation. In previous works such solutions have been tracked for the entire post-buckling regime both numerically and through a Rayleigh-Ritz approach.

A very rich structure is known to exist in the subcritical load range but has been proved to exist for only a certain family of reversible systems. Studies have concentrated on symmetric homoclinic solutions and the asymmetric solutions which bifurcate from these solution paths. The primary solution is known to exist for the entire subcritical parameter range and all other symmetric and associated asymmetric solutions exist strictly for values less than critical. Here we uncover a new family of antisymmetric solutions and some asymptotic analysis suggests that the primary antisymmetric solution exists over the same range as does the primary symmetric solution. A perturbation approach can be used to describe the bifurcation hierarchy for the novel antisymmetric forms. We illustrate a unified approach which is able to predict the circumstances under which non-divergent localized solutions are possible and the results of the analysis are compared with some numerical solutions.

History

Publication title

Proceedings of the Royal Society of London A

Volume

455

Issue

1986

Pagination

2351-2370

ISSN

1364-5021

Department/School

School of Natural Sciences

Publisher

Royal Soc London

Place of publication

6 Carlton House Terrace, London, England, Sw1Y 5Ag

Rights statement

The Royal Society

Repository Status

  • Restricted

Socio-economic Objectives

Expanding knowledge in the mathematical sciences

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