# Point Deflection in Topological Interlocking Plates
## 2 Motion of Blocks in Topological Interlocking Beams/Plates and Simplified Modeling  
In topological interlocking the blocks are not glued to each other and are free to move apart from each other. The resistance to the movement of the blocks is only provided by the stiffness of the peripheral constraint, which is considerably lower than the stiffness of the blocks themselves. This makes it possible to neglect the deformation of the block material. The absence of block binding also gives another mechanical feature of TI assemblies: each block has rotational degrees of freedom. Furthermore, since the blocks are far from being spherical, the relative rotation of neighboring blocks creates forces that push them apart, the phenomenon called elbowing [52], which produces additional deformation of the TI plate. This movement is resisted by the elasticity of the peripheral constraint.  
In order to understand the effect of rotational degrees of freedom and elbowing, consider a 2D model of a cross-section of a TI plate in a co-ordinate frame (\(x\), \(y\)); see the schematics in Figure 5. (This is essentially a toy model presented to investigate the main  
Figure 6: Illustration of localization lines developing in TI cubic assembly under indentation (observed in experiments and numerical modeling [50,51]): (**a**) central section of the TI assembly of cubes; and (**b**) generator cube with a hexagonal central section shown.  
features of the mechanics of the deformation of interlocking structures. It extends the simplification proposed in [22].) Assume each block to be infinitely rigid. Each block can have positive relative displacement, \(\Delta u_{x}>0\), and rotation, \(\Delta\omega_{z}\) (the \(z\)-axis being directed from the reader), with zero vertical relative displacement, \(\Delta u_{y}=0\), ensured by interlocking. (This does not exclude macroscopic vertical deflection, that is \(u_{y}\
eq 0\) allowed by finite stiffness of the peripheral constraint.) Given the in-plane (axial) compression applied by the peripheral constraint, the relative displacement, \(\Delta u_{x}>0\), can only be produced by relative block rotation (elbowing). The relative rotation results in corner interactions of the neighboring blocks. Figure 7 shows a fragment of a topological interlocking beam with equal symmetrical blocks, Figure 7a, or with alternating blocks, Figure 7b, of shapes of two types. (We only consider curved contact surfaces, the ones that make topological interlocking beams possible). It is evident that the relative displacement and rotation under deflection-produced loading are controlled by the action of corners as long as the rotation is sufficient to open the contacts. The corners of these blocks coincide with the corners of simple rectangular blocks (broken lines in Figure 7a,b). Therefore, it is sufficient to model the blocks in a cross-section of interlocking plates by rectangular blocks, with restricted relative vertical displacements representing the effect of topological interlocking; this is expressed by the condition \(\Delta u_{y}=0\). We also note that if \(F_{x}\) is the force (e.g., the constraining force), the moments created by it with respect to the block centroid are the same in all three cases shown in Figure 7.  
We now use a beam-like assembly of rectangular blocks, Figure 7c, with relative vertical block displacement, \(\Delta u_{y}=0\), as a model of the TI beam. The following section uses this model to derive the point nature of beam deflection.