Abstract
We report the results of a numerical and theoretical study of buckling in elastic columns containing a line of holes. Buckling is a common failure mode of elastic columns under compression, found over scales ranging from metres in buildings and aircraft to tens of nanometers in DNA. This failure usually occurs through lateral buckling, described for slender columns by Euler’s theory. When the column is perforated with a regular line of holes, a new buckling mode arises, in which adjacent holes collapse in orthogonal directions. In this paper, we firstly elucidate how this alternate hole buckling mode coexists and interacts with classical Euler buckling modes, using finite-element numerical calculations with bifurcation tracking. We show how the preferred buckling mode is selected by the geometry, and discuss the roles of localized (hole-scale) and global (column-scale) buckling. Secondly, we develop a novel predictive model for the buckling of columns perforated with large holes. This model is derived without arbitrary fitting parameters, and quantitatively predicts the critical strain for buckling. We extend the model to sheets perforated with a regular array of circular holes and use it to provide quantitative predictions of their buckling.
Highlights
Buckling instabilities of elastic structures subjected to deforming forces are found on all scales [1] ranging from large-scale applications such as aircraft to the engineering of DNA [2]
If an elastic sheet is perforated with a two-dimensional square array of circular holes, the sheet can exhibit pattern switching upon compression that internalizes the buckling: the circular holes deform into ellipses with adjacent holes elongated in orthogonal directions [5,6]
Act as a localized buckling mode in which, deformation occurs throughout the length of the column, the critical strain is independent of column length and depends only on the geometry of the holes
Summary
Buckling instabilities of elastic structures subjected to deforming forces are found on all scales [1] ranging from large-scale applications such as aircraft to the engineering of DNA [2]. The Euler buckling of sufficiently long columns with a line of holes is qualitatively similar to that of solid columns, with the critical strain for buckling scaling as the inverse square of the column length (or number of holes), cr ∼ 1/(Nh) (figure 2a–c). As with the sliding regime of the Euler mode, this is a localized buckling mode, characterized by independence of the critical strain on N, and agreement between predictions from finite-length and periodic columns. Comparison of figure 2a and 2b suggests that an increase in h/W promotes the localized sliding regime of the Euler mode This is supported by the results, in which the colours of the crosses indicate the ratio between critical strains of the second to first Euler mode bifurcations; as alternating mode cr2/ cr.
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