Truncated cones from indenting a clamped disk

Abstract

When a simply supported thin disk is indented by a centrally applied point force, it buckles out-of-plane to form a shape dominated by two conical portions: a uniform region indenting against the support, interrupted by a smaller elevated portion detached from the support, altogether known as a “developable cone” or d-Cone. If a central circular region of the disk is clamped instead, then the buckling complexion changes markedly: The indenting region is interspersed with several detached and elevated cones, now “truncated,” where their number depends on the clamping extent as well as the radius of the circular simple support. Studies of d-Cone kinematics often consider its shape as an analogous vertex, which forms by folding along hinge lines separating triangular facets. We extend this methodology by, first, showing that each truncated cone, or “t-Cone,” operates as a pair of connected d-Cone vertices that fold synchronously and that their number, viz. distribution, around the indented disk stems from optimal “packaging” of the folded shape in the annular space between the clamping edge and support; furthermore, because our analysis presumes a geometrically dominant character, it captures the “saturated,” i.e., final number of t-Cones, in experiments from a recent study. Our predictions agree rather well.