# Point Deflection in Topological Interlocking Plates
## 7 Conclusions  
The main feature of topological interlocking (TI) assemblies is the absence of bonding between their elements (blocks), which allows relative block rotation and subsequent elbowing. It is the rotations and elbowing that lead to the principal difference between the behavior of interlocking structures and solids demonstrated by the phenomenon of point deflection and localization. This new concept sets up a framework for future experiments and the design of topological interlocking structures. The proposed concept is based on the consideration that the relative rotations of non-cylindrical or non-spherical blocks lead to elbowing, that is, transferring rotation into in-plane displacement (parting) of the blocks. This has a profound effect on the out-of-plane deformation of TI beams and plates: elbowing increases load in the lateral constraint (e.g., increases tension in the post-tensioning cables). As a result, the bending resistance of the beam/plate is transferred from the conventional bending stiffness, which is zero in the TI assemblies, to the resistance effected by the lateral constraint. This allows the modelingbending of TI beams using the concept of a tensioned string, even if the constraint is external. When point forces are applied, the TI beam undergoes point deflection, whereby the bending is localized at the points of force application with linear deflection in-between. Another prominent consequence of block rotations is the force-deflection relationship containing post-peak softening, which can be interpreted as a manifestation of apparent negative stiffness (total energy of the system TI beam/plate plus the peripheral constraint is positive-definite). The post-peak phase of beam deflection can also be modeled using the concept of tensioning string and elbowing. The post-peak softening phase reflects the unstable phase of the deformation process, which is an important feature of the behavior of TI structures. Damping during the bending of TI beams is produced by the transfer of the energy of oscillations to the kinetic energy of the air squeezed out due to interface closing between neighboring blocks in the process of oscillations. Being less efficient than classical damping, air squeezing damping has significant advantages over classical external damping devices, as air squeezing damping is an integral part of TI structures and hence does not require the installation of additional dampers. This property can be used for designing hybrid materials and materials with internal architectures, where sound absorption/attenuation is a design parameter. The proposed theoretical framework and the models are important for developing methods for controlling the deformation of TI structures and for designing flexible structures with intrinsic energy absorption. Conceptualization, A.V.D. and E.P. ; methodology, E.P. ; formal analysis, A.V.D. ; writing--original draft preparation, A.V.D. ; writing--review and editing, E.P. All authors have read and agreed to the published version of the manuscript. This research received no external funding. Not applicable. The original contributions presented in this study are included in the article. Further inquiries can be directed to the corresponding author. The authors declare no conflicts of interest. Appendix A Equivalence of Constraining by the Post-Tensioning Cable and by the Peripheral Constraint  
Consider two neighboring blocks, which assume under load the relative rotation \(\theta\) (Figure 7c). In the case of constraining by a pre-tensioning cable running through the block centers, the increase in the cable length is as follows:  
\[\Delta l_{c}=2b\sin\frac{\theta}{2} \tag{1}\]  
Similarly, in the case of peripheral constraint, the increase in the cable length is as follows:  
\[\Delta l_{c}=4b\sin(\theta/2) \tag{2}\]  
In the case of constraint by a pre-tensioning cable running through the block centers, the cable shape corresponding to the minimal length increase is the point deflection shown in Figure 8b.