Curvature in Compressed Thin Cylindrical Shells Approaching the Isometric Limit

We study the buckling of a thin cylindrical shell constrained to slide onto an inner non-deformable pipe. Our goal is to characterize the relationship between the shell thickness and the localization of stresses by using curvature measurements. First, we induce surface buckling by immobilizing one end of the shell and applying force to the other end. Then, we obtain a virtual reconstruction of the surface from 3D optical scanning and compute the Gaussian curvature for every point on the mesh. We find that as the shell gets thinner, the distribution of Gaussian curvatures becomes broader. However, surprisingly, the mean of the Gaussian curvature distribution increases. Furthermore, measurements of areas enclosed by the parabolic lines around protruding vertices from the buckled surface show that the transitions between regions of positive and negative Gaussian curvature are more localized. Finally, the Gaussian curvature reveals the formation of substructures within the lobes around the vertices. These results demonstrate that the evolution of the cylindrical shell towards the isometric limit represented by the well-known Yoshimura pattern is non-trivial.