Pinching an open cylindrical shell: Extended deformation and its persistence

Against external loads, shells with ideal symmetries, such as perfect spheres and cylinders, show enhanced rigidities, whose origin is known to be largely geometric and is often termed "geometric rigidity." In contrast, shells with free boundaries (i.e., open shells) may respond quite differently to external loads. They may deflect significantly as free boundaries may allow (almost) inextensional bending deformations that are impossible in the case of closed surfaces. Herein, we investigate the smooth deformation of an open cylindrical shell induced by pinching through a combination of experiment, simulations, and theory. To be more specific, we apply a localized pinch at one end of a long semi-cylindrical shell, and investigate the distance at which the shell recovers its undeformed form; we call this distance the persistence length of a pinch. We establish a new scaling law for the persistence length of shallow shells, which is shown to be valid for an arbitrarily deep shell. We show that the persistence length of a sufficiently deep open cylinder exceeds that of a closed cylinder, thereby highlighting the importance of the subtle interplay between the intrinsic curvature and the free edges on shell mechanics. These findings are rationalized by a detailed asymptotic analysis of the shallow shell equations. Our results reveal how the free boundaries essentially alter the balance between bending and stretching in thin shells.