Convexity induced rigidity transitions

A fundamental theorem in rigidity theory due to Cauchy states that all convex polyhedrons in three dimensions are rigid, i.e. the polyhedron cannot be deformed without changing the shape of at least one of its faces at some energy cost. However, a polygon in two dimensions is floppy irrespective of its convexity and can be deformed with no energy cost. This property is consistent with Maxwell's constraint counting scheme. Our numerical results show that under finite isotropic expansion, an area-conserving polygon rigidifies when it achieves convexity, as does two area-conserving polygons sharing an edge. This demonstrates a link between geometry and mechanics. We finally study 2D spring networks with several edge sharing polygons. We determine the existence of a rigidity-convexity correspondence in the spring network under isotropic expansion.