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Fig. 1. Four buckling modes of axially compressed columns with different width-to-length ratios. (A) A 2D hyperelastic column with width-to-length ratio w/L is subjected to a compressive force F or a displacement delta l that eventually results in buckling of the column. (B) Due to symmetry, only the top half of the column in (A) is selected for finite element simulations, where the bottom side is constrained by a symmetric boundary condition and the mid-point of the top side is restricted to moving vertically to eliminate rigid-body motions. A very small displacement defect delta d in the horizontal direction is introduced to the bottom side of the half column in its stress-free state (light blue) to trigger buckling. A quarter of a circle with a small radius r is introduced as a defect to trigger the initiation of a crease at both the bottom right and top left corners of the half column. A rigid frictionless surface (dashed line) abuts one of the quarter-circle defects to form a self-contacting fold. (C) The relations between normalized compressive force F/(w) and displacement delta l/L for columns with different w/L. Inset: the deformed shape of the columns in post-buckling from the finite element simulations. The color indicates the level of minimum principle logarithmic strain. (D and E) Normalized free energy difference between the buckled state and flat state (with subscript 0) as a function of applied displacement delta l/L (D) and force F/(w) (E). The dashed lines indicate unstable equilibrium path. ((C-E) share the same legend).
This and the next image are from:
Yuzhen Chen and Lihua Jin (Department of Mechanical and Aerospace Engineering, University of California, Los Angeles, CA 90095, USA),
“Snapping-back buckling of wide hyperelastic columns”, Extreme Mechanics Letters, Vol. 34, Article 100600, January 2020, https://doi.org/10.1016/j.eml.2019.100600
ABSTRACT: The mechanical instability of columns with a low width-to-length ratio under axial compression has been studied for more than 260 years, known as the Euler buckling. Such columns buckle at a critical strain on the order of 1%, after which the compressive load continuously increases with the displacement. Recently, in the advance of soft robotics and mechanical metamaterials, researchers have harnessed buckling of high width-to-length ratio columns to achieve new functions. However, buckling and post-buckling of these columns are not well studied. Here we show hyperelastic columns, depending on their width-to-length ratio, can undergo continuous, snapping-through, or snapping-back buckling. In particular, we identify a new snapping-back mode of column buckling, in which beyond the onset of buckling, a column bends to form a sub-critical crease. Our analytical discrete model reveals that snapping-back buckling results from strong coupling between stretching and bending. A phase diagram is constructed to demarcate the different buckling modes of axially compressed columns.
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