# Point Deflection in Topological Interlocking Plates
## 1 Introduction
While for some blocks (e.g., cubes) the removal of a block is observed to lead to the collapse of the whole assembly, other shapes, such as tetrahedra and octahedra, can tolerate the removal of some blocks. In the process of random block destruction, the TI plate of osteomorphic blocks requires the removal of 24.62% of the blocks to collapse [37]. The ability of the blocks to move independently has a strong effect on the way that TI structures oscillate. When the oscillations proceed in the plane of the TI structure (or a planar part thereof), the unbonded block interfaces form what are called bilinear oscillators, that is, oscillators with different stiffnesses in tension and compression, as shown in [38]. Figure 4: Crack growing towards an interface. Singularity of tensile stress, \(\sigma_{x}\), opens the interface, which prevents the crack from further growth (see [27] for details). Figure 3: Two sections of TI assemblies: in the up-constraining section the reference block is prevented from moving up by its neighbors; in the down-constraining section the reference block is prevented from moving down (see also [23]). The reason for this is that when two neighboring blocks in a chain (see Figure 5 as an example) move towards each other (e.g., in the process of oscillations), they resist with half of the stiffness of the blocks, while when the blocks move apart, the maximum resistance is provided by the constraint, which is usually much softer than the blocks. If the blocks are sufficiently stiff, the simplest approximation is given by a so-called impact oscillator, which is an oscillator with infinite stiffness in compressions such that only positive relative block displacements are possible. The main feature of the impact oscillators is the presence of multiple and half resonances [39]. The possibility of damping associated with block-to-block impact with limiting restitution was discussed in [40]. Another important effect of non-bonded interfaces is the high energy absorption exhibited by TI structures, as experiments [41] on sound absorption show. It was also demonstrated in [41] that gluing the blocks together considerably reduces the energy absorption, almost negating it. The damping was also found to decrease with the increase in the pressure exerted by the lateral constraint [42; 43], as the parting of the blocks becomes harder under high constraining stress. Increased energy absorption in dynamic impact is also reported in [7; 44]. A possible mechanism of energy absorption is the transformation of the energy associated with the closure of gaps between the blocks into the kinetic energy of squeezed-out air [45]. This mechanism works only during some phases of oscillations, and hence the damping is discontinuous. The possibility of modeling this type of energy absorption will be discussed in this paper. A common method of testing the deflection in TI plates is to subject them to central point loading (indentation) in the presence of a peripheral frame providing the constraint. Experiments demonstrate that some TI structures show an effect of negative stiffness (post-peak softening) between the applied central displacement and the point force (the corresponding negative energy is of course absorbed by the frame). The negative stiffness was observed in planar TI assemblies of cubes [46; 47], tetrahedra [48], and non-planar assemblies of osteomorphic blocks as well as in assemblies of elements of other shapes [8]. Experiments with central point loading of TI beams and plates show another interesting feature: the deflection is localized at the center in the case of beams [49] or in the straight lines in interlocking plates. (The localization lines originate from the loading point [46; 47; 49; 50]; their directions are controlled by the symmetry of the tessellation of the central plane of the TI plate; see the schematics of experiment [50] in Figure 6.) Similar localization lines were obtained in numerical modeling [51]; however, no mechanism was proposed to explain this phenomenon.