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 will resist exactly three load cases and comply with the other three, provided the shell is simply connected, meaning it has no holes and no handles.

Figures (2)

• Figure 1: The static-geometric analogy for macroscopic fields. Wide arrows are admissible stresses; thin arrows are isometric deformations: (a) in a plane, admissible longitudinal tension corresponds to isometric lateral bending; (b) in a singly corrugated membrane, an admissible longitudinal moment corresponds to isometric lateral extension; (c) in a multiply connected grid of tubes, admissible longitudinal tension has no corresponding isometric lateral bending.
• Figure 2: Minimization of the effective membrane stiffness for discrete shells with (a) $2\times 2$, (b) $4\times 4$, and (c) $6\times 6$ nodes per unit cell. Random initial geometries are in the top row; geometries with optimized elevations are in the bottom row. The trace of the effective membrane stiffness tensor is reduced by a factor of $\sim 2$, of $\sim 10^{10}$ and of $\sim 10^{10}$, respectively.