Giles W. Hunt

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.

The Maxwell stability criterion in pseudo-energy models of kink banding

Kink banding, common to many structures in nature and engineering, has several distinctive features---notably highly nonlinear snap-back instability leading to localization and sequential lock-up. The proposed friction model, although simplified, introduces these defining characteristics without obscuring them by including other effects of lesser immediate significance. In the absence of small imperfections or disturbances, linearized theory suggests that in its pre-kinked configuration the system never goes unstable. However, under sufficient applied end-displacement it is shown to be in a state of extreme metastability, such that micro-disturbances would trigger the nonlinear response. To overcome this problem we adopt an energy approach based on a global rather than a local stability criterion. When applied to imperfect systems with small initial misalignments, the critical displacement thus defined shows little of the sensitivity expected from other stability criteria, and provides a useful lower bound on the expected critical displacement and associated load. We offer the frictional model not as a complete description, but as a prototype to which other components can be added. Thus it provides information on the triggering, localization and lock-up processes, for example, but not on the choice of kink band width which is fixed a priori. Suggestions are provided throughout of extra energy contributions which will extend the models capability.

Homoclinic and heteroclinic orbits underlying the post-buckling of axially-compressed cylindrical shells

Abstract A structural system with an unstable post-buckling response that subsequently restabilizes has the potential to exhibit homoclinic connections from the fundamental equilibrium state to itself over a range of loads, and heteroclinic connections between fundamental and periodic equilibrium states over a different (smaller) range of loads. It is argued that such equilibrium configurations are important in the interpretation of observed behaviour, and govern the minimum possible post-buckling loads. To illustrate this, the classical problem of a long thin axially-compressed cylindrical shell is revisited from three different perspectives: asymptotic conjecture, analogy with nonlinear dynamics, and numerical continuation analysis of a partial spectral decomposition of the underlying equilibrium equations. The nonlinear dynamics analogy demonstrates that the structure of the heteroclinic connections is more complicated than that indicated by the asymptotics: this is confirmed by the numerics. However, when the asymptotic portrayal is compared to the numerics, it turns out to be surprisingly accurate in its Maxwell-load prediction of the practically-significant first minimum to appear in the post-buckling regime.

Interactively induced localized buckling in sandwich structures with core orthotropy

Recent work on the interactive buckling behavior of compressed sandwich structures (Hunt and Wadee, 1998) is extended to orthotropic core materials. The variation of the material properties can tend to maximize the interactive effect such that overall and secondary localized modes are triggered almost simultaneously, giving rise to highly unstable post-buckling behavior in systems of practical dimensions.

Computation of localized post buckling in long axially compressed cylindrical shells

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

Force-chain buckling in granular media: a structural mechanics perspective

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

Arnold tongue predictions of secondary buckling in thin elastic plates

The stability of post-buckled states for simply-supported flat elastic plates under compression is investigated for a range of in-plane boundary conditions. The von Karman plate equations are reduced to a series of ODEs which are solved numerically under parametric variation of both load and length. Results are checked against full numerical solutions of the PDEs, and comparison with a modal analysis highlights the dominant passive contaminations. The nondimensional amplitude at secondary bifurcation, for any combination of modes and all plate lengths, is presented in a concise form using the parameter space of Arnold tongues. This demonstrates that compound bifurcation represents a worst case for post-buckling reserve, and that long plates have inherently more such reserve than short plates. It is also shown that stiffening the boundaries against in-plane movement is destabilizing, in that it induces mode jumping at secondary bifurcation to occur at an earlier stage in the post-buckling regime.

Mathematical modeling of the radial profile of basilar membrane vibrations in the inner ear

Motivated by recent experimental results, an explanation is sought for the asymmetry in the radial profile of basilar membrane vibrations in the inner ear. A sequence of one-dimensional beam models is studied which take into account variations in the bending stiffness of the basilar membrane as well as the potential presence of structural hinges. The results suggest that the main cause of asymmetry is likely to be differences between the boundary conditions at the two extremes of the basilar membranes width. This has fundamental implications for more detailed numerical simulations of the entire cochlea.

Multi-layered folding with voids.

In the deformation of layered materials such as geological strata, or stacks of paper, mechanical properties compete with the geometry of layering. Smooth, rounded corners lead to voids between the layers, while close packing of the layers results in geometrically induced curvature singularities. When voids are penalized by external pressure, the system is forced to trade off these competing effects, leading to sometimes striking periodic patterns. In this paper, we construct a simple model of geometrically nonlinear multi-layered structures under axial loading and pressure confinement, with non-interpenetration conditions separating the layers. Energy minimizers are characterized as solutions of a set of fourth-order nonlinear differential equations with contact-force Lagrange multipliers, or equivalently of a fourth-order free-boundary problem. We numerically investigate the solutions of this free-boundary problem and compare them with the periodic solutions observed experimentally.

Interactive Bending Behavior of Sandwich Beams

A multi-degree-of-freedom nonlinear Rayleigh-Ritz formulation for a sandwich beam is developed and is used to demonstrate the possible sudden destabilizing effects associated with wrinkling on the compressive face. Through-core stretching and core shearing effects are included. Nonlinear load-deflection curves, for various loading conditions including point loads and uniformly distributed loads, are presented