Eversion Buckling of Toroidal Shells: Mechanism, Bifurcation, and Applications in Energy Absorption

Abstract

Thin shells can store and release elastic energy through large geometric transformations, but their post-deformation stability is often highly sensitive to geometry and loading direction. Here, we study the eversion of open toroidal shells generated by revolving a meridional arc about an external axis and identify a geometry-controlled instability, which we define as eversion buckling. After eversion, the shell may either remain in an axisymmetric everted state or spontaneously collapse into a non-axisymmetric configuration. We show that this transition is governed by the competition between bending and membrane energies. A scaling analysis leads to a dimensionless geometric parameter that predicts the onset of eversion buckling, namely, the spontaneous loss of stability of the axisymmetric everted configuration after unloading. Finite element simulations and experiments confirm that this parameter collapses the stability data for shells with different geometries, including circular and semi-elliptical generating curves. The loss of axisymmetric stability is consistent with a pitchfork-like symmetry-breaking transition, in which the collapsed state has no preferred in-plane direction because of the axisymmetry of the everted shell. When the axisymmetric everted configuration is stable, a finite perturbation can trigger rapid snap-through toward a lower-energy collapsed equilibrium, accompanied by substantial volume contraction and a triggering response that is relatively insensitive to boundary constraints. We further show that assemblies of these shells behave as granular energy-absorbing systems, exhibiting a stable stress plateau, delayed densification, and substantial frictional dissipation during collective rearrangement. These results provide a mechanics-based route for designing shell assemblies with robust, direction-insensitive energy absorption.