Geometric and mechanical rules governing the confined coiling of thin shells

The confined coiling of a thin shell appears in both natural morphologies and engineering designs. Yet its underlying geometric and mechanical principles remain unclear. Hence, we investigate how a tape spring, a representative of thin shells, coils around a rigid cylindrical hub under a tension. Combining experiments, simulations, and theoretical analysis, we find that the shell consistently adopts a regular polygonal configuration featuring the periodic localized folds. This discrete folding pattern arises as the shell curvature prevents a smooth coiling, driving it into a symmetric and periodic arrangement of folds. We show that this pattern emerges from a fundamental interplay between geometric incompatibility and energy minimization. Applying the principle of virtual work, we establish a quantitative relation between the applied tension and the number of folds. The above results uncover the geometric and mechanical rules governing the coiling of thin shells, providing a general framework for understanding and controlling folded coiling in curved structures.