Sliding of a cylindrical shell into a rigid hole

Fitting two different materials is a versatile methodology in manufacturing complex structures. One of the canonical models for fitting is the snap-fit model, in which flexible materials and rigid structures are assembled by pushing their interlocking components together. The assembly via snap-fit is often accompanied by large deformations of flexible structures and abrupt force drops, highlighting the role of elasticity, geometry, and contact friction. Despite several model studies revealing fundamental mechanics for snap-fit, the current snap-fit design relies on prototyping and empirical rules. In this paper, we analyze a snap-fit model in which a naturally curved beam slips into a rigid hole. We construct an analytical model based on the theory of elastica with contact friction and demonstrate that its predictions are in excellent quantitative agreement with both simulations and experiments. We find three distinct sliding modes: folding, pinning, and unfolding. The classification is systematically organized in a phase diagram based on the geometric parameters of the shells and the hole. Our study complements existing approaches by providing a predictive framework for contact-based structures that involve friction, elasticity, and geometry, and sheds light on a unified understanding of the interactions between an elastic and a rigid body.