# Point Deflection in Topological Interlocking Plates
## 1 Introduction  
Since the introduction of the concept of topological interlocking (TI) [1, 2, 3, 4], considerable effort has been directed towards describing the mechanisms of deformation and failure in TI structures. The TI elements (blocks) are not bonded to each other but rather hold kinematically owing to the special geometry, which only requires the presence of peripheral constraint to maintain structural integrity. It was shown that all platonic shapes (tetrahedron, cube, octahedron, dodecahedron, and icosahedron) (Figure 1), as well astheir truncations, including the buckyballs, permit topological interlocking into plate-type structures [1; 2; 3; 4; 5]. Additionally, a special shape, an osteomorphic block with convex/concave surfaces (Figure 2), was introduced [6], allowing for the assembling of interlocking structures containing corners and columns. Interlocking shapes were further analyzed in [7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18]. Methods of designing topological interlocking structures were proposed in [19; 20; 21; 22]. The main idea of topological interlocking is illustrated in Figure 3, where functional sections are considered. The reference (central) block is prevented from moving up by the neighboring blocks in the up-constraining section, while moving down is prevented by the neighboring blocks in the down-constraining section. The sections can be parallel, smoothly transferring from up-constraining to down-constraining; this will produce osteomorphic blocks. (A similar representation of interaction of TI blocks is used in [23].) The sections can also be perpendicular to each other (TI assemblies of tetrahedra), at \(120^{\circ}\) to each other (TI assemblies of cubes, octahedra, and dodecahedra), or at other angles. The traces of the  
Figure 1: TI assemblies of (**a**) tetrahedra; (**b**) cubes; (**c**) octahedra; (**d**) dodecahedra, packing 1; (**e**) dodecahedra, packing 2; and (**f**) octahedra; the figures are drawn following [1; 2; 3; 4; 5]. Figure 2: TI assemblies of osteomorphic blocks [6]: (**a**) planar assembly; (**b**) corner assembly. blocks in the section do not have to be straight: any curves are possible. This principle can also be used for developing new interlocking shapes. As a generalization of the concept of topological interlocking, hierarchical interlocking was introduced in [24], which covers a range of scales with different interlocking shapes. It should be noted that the proposed small-scale part of topological interlocking resembles the sliding resisting friction between two blocks with asperities on the contacts; their interaction leads to friction and dilation [25]. Conversely, the mechanism of the TI structural stability can be thought of as being a representation of large-scale friction. The main feature of TI structures is the absence of bonding, resulting in the high compliance [25, 26] and flexibility of TI structures. The presence of non-bonded interfaces leads to elevated fracture growth resistance [6]. This is due to the fact that the tip of a growing fracture has singularity in all stress components. In particular, the singularity of tensile stress acting on the continuation of the fracture plane, \(\sigma_{y}\) (in the coordinate system shown in Figure 4), which provides the mechanism of Mode I crack growth, is the same as the singularity of tensile stresses, \(\sigma_{x}=\sigma_{y}\), acting on the planes normal to the direction of fracture propagation. This second stress singularity opens the inter-block boundaries as the fracture approaches them (Figure 4), which eventually arrests the fracture growth [27]. Based on these features, various engineering applications of TI structures were proposed, e.g., those in [7, 28, 29, 30], as well as including marine and coastal structures [31] and new types of materials [32, 33, 34, 35, 36]. The reliance on the neighboring blocks to maintain structural integrity seems to make the TI structures very sensitive to the destruction (elimination) of individual blocks.