Stability of split structures: degeneracy breaking and the role of coupling

Abstract

The present work is inspired by the need to understand the elastic stability of a class of structures that appear in a variety of seemingly unrelated fields. Here we consider several problems involving the stability of two or more slender structures coupled at the ends. In a sequence, we consider a bilayer beam, then a multilayer split beam, chains of elastically coupled rigid rods, a plate with a symmetric cut out, and finally several plate strips elastically coupled. We also study the instability of a biological structure known as the mitotic spindle. We report cooperative, competitive, and antisymmetric buckling of the bilayer split beam; and their dependence on the geometric parameters. Then we identify the mechanisms of elastic deformation, including additional strain induced by the misfit of two layers tied together at ends, that explains the observed behaviour. This is extended to buckling of a multilayer structure, i.e. a stack of thin elastic layers coupled at the ends. We also report rapid decay of the buckling amplitude of layers along the stacking direction, observed in simple experiments. We theoretically study a chain of elastically coupled rigid rods as the simplest model of this behaviour and report that coupled identical members, in the absence of any disorder, show spatially extended buckling modes, i.e. buckling amplitudes are periodically modulated. Analogies are drawn with a physically unrelated, yet mathematically close problem of wave propagation in periodic media. Introduction of irregularity leads to the spatial exponential decay of the amplitudes, i.e. localisation of buckling modes and thus associated Lyapunov exponents. We show that the strength of buckling localisation depends on the coupling-to-disorder ratio. Next, we study the instability of rectangular plates with one or more cut outs placed periodically. The first problem reveals two types of buckling modes – in-phase buckling and out-of-phase buckling of the two elastically coupled plate strips. Energy contributions from cylindrical bending and twist of the coupling region drive the structure from degeneracy to where the mode character changes. The second problem of multiple strips elastically connected reveals that the in-phase and out-of-phase modes become periodically modulated and the respective buckling loads appear in clusters. If the structure is perfectly ordered, the entire clusters of buckling loads are inverted in the degeneracy point via N-fold crossing. Infinitesimally small disorder triggers repulsion of eigenvalues and strong localisation occurs. We characterise this e↵ect comprehensively by calculating Lyapunov localisation factors and report regions of structural parameters for which high and moderate sensitivity to disorder is observed. Finally, mitotic spindles were studied using continuum modelling of the slender bio-structures also accounting for the interaction with the environment of the cell. Interesting buckling modes with spatial features such as coupled bending and torsion of filaments were observed