Mk Wadee

Comparative lagrangian formulations for localized buckling

An energy functional for a strut on a 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, and carry some quite general features. In each case, the kinetic energy is an indefinite quadratic form. In fast space, this leads to an escape phenomenon with fractal properties. In slow space, kinetic energy is added to a potential contribution that is familiar from modal formulations. Together, and in conjunction with a recent set of numerical experiments, they illustrate the extra complexities of localized, as opposed to distributed periodic, buckling.

Asymptotic and Rayleigh–Ritz routes to localized buckling solutions in an elastic instability problem

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

Localized Elasticae for the Strut on the Linear Foundation

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

EFFECTS OF EXPONENTIALLY SMALL TERMS IN THE PERTURBATION APPROACH TO LOCALIZED BUCKLING

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 so as 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.

Galerkin approximations to static and dynamic localization problems

Abstract A new Galerkin-type procedure is established which, unlike the classical approach, does not rely on the final shape being composed of linearly independent modes. The procedure is applied to the evolution of a localized buckle of a thin elastic strip within a visco-elastic medium. Unlike the related elastic problem, no clear-cut linear eigenvalues exist to model wavelength and exponential growth/decay in the tails of the buckle pattern. The new procedure introduces variables to measure these effects, and allows them to change in time. This results in a more natural evolutionary process than with fixed mode shapes. Analysis is run within an algebraic manipulator (M aple ) and checked against that of a numerical boundary-value solver (COLPAR).

Elasto-plastic localised responses in one-dimensional structural models

This work complements recent developments concerning the buckling of beams lying on a nonlinear (non-convex) elastic foundation, and also reports on some investigations on the role of material nonlinearity. Two structural models are studied using a simple elasto-plastic constitutive relationship, and buckling problems are formulated as reversible fourth-order differential equations. It is demonstrated that modulated responses are possible under certain circumstances. Some numerical simulations are presented supporting the analytical findings.

Fat-fractal scaling in an area-preserving structural mapping

Abstract The chaotic nature of the trajectories in a mapping related to a discrete structural link model is quantified using the numerical techniques of fractal analysis. The results presented are for localized buckling (homoclinic) solutions for a rigid link model supported by linear-elastic springs, which has been shown to be of importance to problems in elastic instability analysis. This type of study complements work which has already been done in the structural domain of an analytical nature. It demonstrates that localized solutions form part of a chaotic regime. The nature of the phase-space maps presented herein is found to be that of a fat fractal which has a definite non-zero area (or measure) unlike, for example, the Cantor set which has zero measure.

On the interaction of uni-directional and bi-directional buckling of a plate supported by an elastic foundation

A thin flat rectangular plate supported on its edges and subjected to in-plane loading exhibits stable post-buckling behaviour. However, the introduction of a nonlinear (softening) elastic foundation may cause the response to become unstable. Here, the post-buckling of such a structure is investigated and several important phenomena are identified, including the transition of patterns from stripes to spots and back again. The interaction between these forms is of importance for understanding the possible post-buckling behaviours of this structural system. In addition, both periodic and some localized responses are found to exist as the dimensions of the plate are increased and this becomes relevant when the characteristic wavelengths of the buckle pattern are small compared with the size of the plate. Potential applications of the model range from macroscopic industrial manufacturing of structural elements to the understanding of micro- and nanoscale deformations in materials.

The interactive bending wrinkling behaviour of inflated beams

A model is proposed based on a Fourier series method to analyse the interactive bending wrinkling behaviour of inflated beams. The whole wrinkling evolution is tracked and divided into three stages by identifying the bifurcations of the equilibrium path. The critical wrinkling and failure moments of the inflated beam can then be predicted. The global–local interactive buckling pattern is elucidated by the proposed theoretical model and also verified by non-contact experimental tests. The effects of geometric parameters, internal pressure and boundary conditions on the buckling of inflated beams are investigated finally. The results reveal that the interactive buckling characteristics of an inflated beam under bending are more sensitive to the dimensions of the structure and boundary conditions. We find that for beams which are simply supported at both ends or clamped and simply supported, boundary conditions may prevent the wrinkling formation. The results provide significant support for our understanding of the bending wrinkling behaviour of inflated beams.