A continuum description of the buckling of a line of spheres in a transverse harmonic confining potential

Abstract

A line of contacting hard spheres, placed in a transverse confining potential, buckles under compression or when tilted away from the horizontal, once a critical tilt angle is exceeded. This interesting nonlinear problem is enriched by the combined application of both compression and tilt. In a continuous formulation, the profile of transverse sphere displacement is well described by numerical solutions of a second-order differential equation (provided that buckling is not of large amplitude). Here we provide a detailed discussion of these solutions, which are approximated by analytic expressions in terms of Jacobi, Whittaker and Airy functions. The analysis in terms of Whittaker functions yields an exact result for the critical tilt for buckling without compression.

Packing soft spheres: experimental demonstrations with hydrogels

“…Our final experiment is related to recent experimental and theoretical studies concerning the buckling of a line of contacting spheres, under compression [49][50][51][52]. We have placed 10 hydrogel beads in a horizontally placed trough made from a cylinder that was cut in half longitudinally.…”

“…A further increase of sphere size makes the buckling profile more prominent. Its shape, shown in figure 8(d), can be interpreted using a theoretical model recently developed by us [50,52]. It concerns the arrangements of contacting hard spheres in a transverse harmonic potential, provided by the transverse profile of the trough in the experiment [49,52].…”