Axisymmetric ridges and circumferential buckling of indented shells of revolution

When poking a thin shell-like structure, like a plastic water bottle, experience shows that an initial axisymmetric dimple forms around the indentation point. The ridge of this dimple, with increasing indentation, eventually buckles into a polygonal shape. The polygon order generally continues to increase with further indentation. In the case of spherical shells, both the underlying axisymmetric deformation and the buckling evolution have been studied in detail. However, little is known about the behavior of general geometries. In this work we describe the geometrical and mechanical features of the axisymmetric ridge that forms in indented general shells of revolution with non-negative Gaussian curvature and the conditions for circumferential buckling of this ridge. We show that under the assumption of "mirror buckling,"a single unified description of this ridge can be written if the problem is nondimensionalized using the local slope of the undeformed shell midprofile at the ridge radial location. However, in dimensional form the ridge properties evolve in quite different ways for different midprofiles. Focusing on the indentation of shallow shells of revolution with constant Gaussian curvature, we use our theoretical framework to study the properties of the ridge at the circumferential buckling threshold and evaluate the validity of the mirror buckling assumption against a linear stability analysis on the shallow shell equations, showing very good agreement. Our results highlight that circumferential buckling in indented thin shells is controlled by a complex interplay between the geometry and the stress state in the ridge. The results of our study will provide greater insight into the mechanics of thin shells. This could enable indentation to be used as a means to measure the mechanical properties of a wide range of shell geometries or used to design shells with specific mechanical behaviors.