A topological counting rule for shells

Abstract

Holding a shell in their hands, one can apply six loads: three by pulling and shearing, and three by bending and twisting. Here, it is shown that the shell resists exactly three load cases and comply with the other three, provided the shell is simply connected, meaning it has no holes and no handles. Formally, it is shown that the space of homogeneous membrane and bending strains, defined in the sense of plate theory, that can be relaxed into an infinitesimal isometry by a periodic, or a statistically homogeneous, deflection is three-dimensional for any simply-connected periodic, or statistically homogeneous, shell, be it corrugated, creased or wrinkled.