Buckling of a tilted line of confined hard spheres

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

“…In the absence of tilt, the discrete equations were also reformulated as a second-order differential equation [2,23,25] for which it is convenient to use as dependent variable ϕ(u), where fðuÞ ¼ tan uðuÞ:…”

“…In the absence of tilt, the discrete equations were also reformulated as a second-order differential equation [2,23,25] for which it is convenient to use as dependent variable ϕ ( u ), whereϕfalse(ufalse)=tan⁡θfalse(ufalse).The variable u replaces the index n in the discrete formulation. In appendix B, we derive the differential equation in the presence of tilt asϕ″false(ufalse)=−4ϕfalse(ufalse)+ϕ(u)G(u)1+ϕ2false(ufalse);see also equation (B 5).…”

“…The smooth modulation of the displacement profile that is generally found for large N (and small compression) invites the (a) Photograph of a buckled line of N = 10 metal spheres (ball bearings) resting in a horizontal cylinder, between two stoppers. In such an experiment, the curvature of the cylinder provides an approximately harmonic potential, that is, a restoring force acting on transversely displaced spheres [1,2]. The angles θ n between contacting spheres and the cylinder axis vary along the line.…”

“…In earlier papers [1,2,[22][23][24][25], we engaged in a numerical search for solutions, which were gathered up in detailed bifurcation diagrams ( plots of energy, or another property, as a function of a control parameter, usually compression). This was done for up to N = 20 spheres [22].…”

“…Displacement of a sphere by a distance R n away from the central axis results in a transverse restoring force f n with magnitude k R n , where k is a force constant.In experiments with hard spheres, placed at the bottom of a horizontal cylinder, the restoring transverse force is provided by gravity, as a transversely displaced sphere experiences the curvature of the cylinder. It can be shown that for the magnitudes of transverse displacements encountered in such a set-up, the restoring force is approximately linear in this displacement, with a force constant k given by k = 2 mg /( d − D ) [2]. Here m and D are, respectively, sphere mass and diameter, d is the cylinder diameter, and g is acceleration due to gravity.We introduce non-dimensional quantities by defining r n = R n / D , where D is the sphere diameter.…”