Rigidity Criteria for Chainmail Consisting of Tessellations of Torus Knots

Interlocked and polycatenated material systems, consisting of discrete, nonconvex particles linked to their nearest neighbors, such as chainmail fabrics, have been shown to undergo a jamming transition that increases their rigidity under boundary compression. This rigidity transition is associated with an increase in contact number between particles. In architected materials, rigidity is described by theories such as the Maxwell criterion. In this Letter, we propose a rigidity theory for a type of interlocked material system: the torus knot tessellation. Torus knot tessellations are structured fabrics composed of particles shaped as torus knots. In these fabrics, we theoretically demonstrate that in-plane rigidity is governed by a modified Maxwell criterion, while out-of-plane rigidity is governed by a crease line criterion. These theories provide a framework for the design of rigidity of these fabrics.