The localization of buckle patterns in elastic structures is reviewed from three complementary viewpoints: ( a ) modal perspective, b formulation which allows the amplitude to modulate an asymptotically defined ‘slow’ space and c dynamical analogy phase suggested by form underlying differential equation. A simple strut on (asymmetric) nonlinear foundation provides typical illustrative example. approaches emphasize different features phenomenon. view illustrates distinctive effects boundary...

Compressed sandwich structures, comprising two stiff face plates separated by a softer core material, while designed principally as efficient integral can lose this quality when faces buckle locally. Interaction between overall (Euler) buckling and local of one suggests that failure will localize into the centre. A variational formulation, leading to pair nonlinear differential equations subject constraints, describes post–buckling response. These are solved combination numerical shooting...

Compression After Impact (CAI) strength is critical to the safety and weight of carbon fibre aircraft. In this paper, standard aerospace industry practice using separate analyses tests for panel buckling CAI challenged. Composite panels with a range stacking sequences were artificially delaminated subject compression testing in fixture that allowed local sublaminate global modes interact. Compared without delamination, interaction reduced strains by up 29%. Similarly, compared restrained...

A hypothesis for the prediction of circumferential wavenumber buckling thin axially-compressed cylindrical shell is presented, based on addition a length effect to classical (Koiter circle) critical load result. Checks against physical and numerical experiments, both by direct comparison wavenumbers via scaling law, provide strong evidence that correct.

An energy functional for a strut on nonlinear softening foundation is worked into two different lagrangian forms, in fast and slow space respectively. The developments originate independently of the underlying differential equation, carry some quite general features. In each case, kinetic an indefinite quadratic form. space, this leads to escape phenomenon with fractal properties. added potential contribution that familiar from modal formulations. Together, conjunction recent set numerical...

A progressive destabilization in compressed elastic structures of sandwich construction is identified, whereby a triggering bifurcation into an overall mode buckling rapidly followed by secondary unstable combination at least two local modes. six degree-of-freedom analysis for strut provided, which tracks equilibrium paths perfect system, specifically pinpointing states bifurcation. The analytical model allows the independent variation bending and shear section, components that are found...

This paper examines the thesis that a process of structural optimization leads inevitably to designs which exhibit notorious failure characteristics often associated with buckling thin elastic shells. means an idealized perfect structure exhibits unstable and compound branching point would fail by explosive instability while nominally real structures containing inevitable small imperfections at scattered loads can be quite considerably lower than idealization. It is shown via fairly wide...

A synthesis of recent progress is presented on a topic that lies at the heart both structural engineering and nonlinear science. The emphasis thin elastic structures lose stability subcritically --- without nearby stable post-buckled state canonical example being uniformly axially-loaded cylindrical shell. Such are hard to design certify because imperfections or shocks trigger buckling loads well below threshold linear stability. resurgence interest in instability phenomena suggests...

Two complementary approximate techniques are developed to describe the subcritical (localized) deflection patterns of elastic struts resting on foundations. One is a double–scale perturbation approach directly from total potential energy functional; other an extension traditional Rayleigh–Ritz analysis. Both make extensive use modern symbolic computation tools and validated against accurate independent numerical solutions. The asymptotic perturbationapproach shows most accuracy at loads...

Parallels are drawn between the response of a discrete strut on linear elastic foundation and force-chain buckling in constrained granular medium. Both systems buckle initially into periodic shapes, with wavelengths that depend relative resistances to lateral displacement, curvature buckled shape. Under increasing end shortening, classical structural model evolves localized form extending over finite number contributing links. By analogy, it is conjectured might follow much same evolutionary...

Localized solutions, for the classical problem of nonlinear strut (elastica) on linear elastic foundation, are predicted from double-scale analysis, and confirmed volume-preserving Runge-Kutta runs. The dynamical phase-space analogy introduces a spatial Lagrangian function, valid over initial post-buckling range, with kinetic potential energy components. indefinite quadratic form admits unbounded corresponding to escape well. Numerical experimentation demonstrates that there is fractal edge...

The general theory of elastic stability is extended to include the imperfection-sensitivity twofold compound branching points with symmetry potential function in one critical modes (semi-symmetric bifurcation). Three very different forms can result, so a subclassification into monoclinal, anticlinal and homeoclinal semi-symmetric introduced. Relating this bifurcation René Thom’s catastrophe theory, it found that point generates an elliptic umbilic catastrophe, while monoclinal lead differing...

Buckling is investigated of a long thin cylindrical shell under longitudinal compression as modelled by the von Kármán–Donnell equations. Evidence reviewed for buckling being localized to some portion axial length. In accordance with this observed behaviour equations are first approximated circumferentially Galerkin procedure, whereupon cross–symmetric homoclinic solutions resulting system ordinary differential sought in direction. Results compared experimental and other numerical data....