Mortarless structures based on topological interlocking

Abstract

We review the principle of topological interlocking and analyze the properties of the mortarless structures whose design is based on this principle. We concentrate on structures built of osteomorphic blocks – the blocks possessing specially engineered contact surfaces allowing assembling various 2D and 3D structures. These structures are easy to build and can be made demountable. They are flexible, resistant to macroscopic fractures and tolerant to missing blocks. The blocks are kept in place without keys or connectors that are the weakest elements of the conventional interlocking structures. The overall structural integrity of these structures depends on the force imposed by peripheral constraint. The peripheral constraint can be provided in various ways: by an external frame or features of site topography, internal pre-stressed cables/tendons, or self-weight and is a necessary auxiliary element of the structure. The constraining force also determines the degree of delamination developing between the blocks due to bending and thus controls the overall flexibility of the structure thus becoming a new design parameter.

Keywords

topological interlocking / fragmented structures / segmented structures / constraint / delamination / bending stiffness

Introduction

Modern construction techniques are essentially based on the centuries-old method of assembling a structure from building blocks strongly connected to each other. It underlies both the ubiquitous brick-and-mortar construction and that with the reinforced concrete blocks or panels whose reinforcement rods are welded together. Even the existing mortarless structures based on conventional interlocking bricks (e.g., [1-5]) rely on various keys and connectors to keep the blocks together.

The main role mortar or connectors play is to prevent sliding of the building blocks relative to each other. Furthermore, given the usual low tensile strength of mortar and connectors the resistance to tensile stresses (created for instance by bending) is primarily ensured by the weight of the structure aided in some cases by the arch effect [6]. At the same time mortar and connectors make the structure stiffer which reduces its resilience to vibrations and seismicity. Indeed, the known mortarless structures whose blocks have some degree of free relative movement, such as dry stone walls [7] and the classical temples (e.g., [8]), have survived for centuries even in seismic zones. Obviously, it is the ability of the blocks to move relative to each other and, in the process, dissipate the vibration energy, which underpins the longevity of these structures. However, the only factors that hold the blocks in place are weight and friction.

Recently, the principle of topological interlocking was introduced [9-22] with a potential to build mortarless structures whose elements are prevented from being fully extractable from the structure, while allowing some degree of relative movement. Topological interlocking is a method to design the special shapes of the building blocks and the corresponding topologies of their assemblies such that no internal block can be removed from the assembly held by kinematic constraints imposed by its neighbors. (In Goodman and Shi’s [23] terminology, these are structures without key blocks.) Only the blocks on the periphery of the structure need to be constrained independently. In construction engineering this can be accomplished either by the weight of surcharge (or their own weight) or by constraining frames, pre-tensioned tendons or cables.

Topological interlocking, while providing structural integrity, permits some restricted movement of the blocks relative to each other. This ensures that cracking does not occur during high amplitude vibrations (e.g., seismic vibrations) and that the vibration energy is dissipated due to friction between the moving surfaces.

The shape of blocks allowing topological interlocking include those of convex polyhedra, such as all five platonic shapes (tetrahedron, cube, octahedron, dodecahedron and icosahedron; see 2 examples in Fig. 1) and their truncated derivatives (including buckyballs) [12,17,21] as well as blocks with specially engineered shapes, such as osteomorphic blocks having two concavo-convex surfaces [14], Fig. 2. The latter are especially attractive as they allow assembling 2D and 3D structures without gaps, Fig. 3. (While the interlocking of blocks of tetrahedral shape and their truncated derivatives, as well as the shape of osteomorphic block were proposed before [24,25], the principle of topological interlocking provided a systematic method of designing new interlocking shapes.) In what follows, our focus will be on the structures that can be built from osteomorphic blocks.

Structures from osteomorphic blocks

Types of structures and their properties

Osteomorphic blocks can be assembled to three basic types of structures: a plate-like assembly, a corner structure and a column, Fig. 3. These can be used to build all structural elements required in construction of a building, including walls, roofs and foundations. The properties of these structures, which define their possible applications, are discussed below.

Fracture resistance

The interfaces between the blocks can present obstacles to the fracture propagation [10,11,26]. Depending upon the magnitude of compression created by the constraint and the amount of friction at the interface, crack offset, Fig. 4, may occur (see details in [27]). This increases the length and waviness of the crack trajectory, which additionally dissipates the energy required for fracture propagation. As a result, the cracking gets localized within separate blocks, rather than developing to a catastrophic crack.

Resistance to block removal

Topological interlocking provides high resistance to removal of the blocks. Furthermore, the interlocking plate-like assemblies of osteomorphic blocks are robust with respect to removal or full destruction of a number of blocks. Indeed, an assembly of osteomorphic blocks retains its load bearing capacity when some blocks are missing [11]. Computer simulations [19] showed that under random failure of osteomorphic blocks the structure resists collapse by percolation of damage until nearly 25% of the blocks fail.

Flexibility

Interlocking plates offer high flexibility over solid plates or plate-like structures made of brick and mortar [11]. This is another consequence of the absence of the binder phase. Increased bending compliance results from two factors: 1) the reduction in the in-plate elastic moduli associated with the presence of interfaces and 2) reduced resistance to bending caused by partial loss of contact at the interface during bending. We call this phenomenon delamination; its model is considered in the following section.

Mobility and Reparability

Topological interlocking offers a unique possibility of erecting demountable and hence movable structures. Indeed, once the constraint is removed, the whole structure can be disassembled, moved to another place and re-assembled. Naturally, after disassembling all damaged blocks can easily be replaced or repaired. It should be noted, though, that repair of a block involves disassembling of part of the structure required to access the block. Alternatively, it may be possible to develop a procedure of repairing damaged blocks in situ, for instance by using special glues and fillers.

Applications

The plate-like assemblies of topologically interlocked blocks can be used to construct flexible foundations that are insensitive to local reductions of the load bearing capacity of the ground and thus efficiently spread the load from the structure. In addition, being inherently segmented such foundations are permeable, which provides channels for dissipating excess pore pressure and thus reduce the risk of liquefaction. This feature, together with flexibility and high fracture resistance, can be important for seismicity-proof construction.

Similarly, interlocking bricks can be used in pavements, especially in the cases when local settlement is to be prevented. Pavements based on osteomorphic blocks are straightforward as the exposed surfaces of the blocks are flat. Tetrahedral, cubic or octahedral blocks will have to be truncated to provide flat working surface for use in pavements. (This method was first proposed in [25], see also [28]).

Another possible application of topological interlocking is in tilings, both protective and decorative. A particularly attractive possibility is to use such tilings in heat shields [15], cladding of furnaces and other similar applications. The main advantage of tilings of this type is an extended service time, since interlocked tiles can only be removed when completely destroyed.

Constraint systems

Structures built from osteomorphic blocks only require unidirectional constraint in the direction crossing the curved contact faces of the blocks. Figure 5 shows possible types of constraint. The simplest one is through self-weight of the blocks or through a surcharge such as the beam shown in Fig. 5. The constraint of this type can be further reinforced by external tension cables. A particular type of constraint by external frame is given when the interlocking structure is built-in as part of a conventional structure. In this case the encompassing structure plays a role of the frame. In some cases it is possible to employ the natural constraint utilizing elements of topography of the terrain or artificial pits. In this case the interlocking layer is placed in a natural or artificial pit such that the constraint is provided by the walls of the pit [18]. The constraining pressure is relatively low in this case. This type of constraint is suitable for foundations, tunnel and excavation lining and, in some cases, for pavements.

Another, and probably more versatile, type of constraint is through internal pre-tensioned cables or tendons. This type of constraint requires interlockable blocks with channels in them.

In principle, the types of constraint outlined above are suitable for all interlocking structures and most applications.

Manufacturing of osteomorphic blocks

Topological interlocking relies upon special shapes and arrangements of the blocks and for that reason is independent of the scale and material of the blocks. The only requirements on the material are sufficient strength to withstand the stress concentrations created, in the case of osteomorphic blocks, by the curved contacting surfaces [29], application-specific functional properties, and the economy of manufacturing. Concrete would be a convenient material to manufacture the blocks by casting in a mold. Given the curved faces of osteomorphic blocks there are two particular requirements. First, the concrete should use aggregates fine enough to fill the sharp corners of the mold. Secondly, the mold should consist of two halves each containing only one curved surface such that the cast block can be easily removed after the mold is split open into the two halves. This leads to the design shown in Fig. 6 where the top view is sketched. The mold is filled from the top and may be closed by a lid. After the block has solidified, the lid is removed, the mold opened and the cast block removed.

Mechanics of interlocking structures

This section introduces the specific features of mechanical behavior of structures based on topological interlocking, which are not present in conventional brick and mortar structures.

Resistance to block removal and the effect on bearing capacity

Resistance to block removal provided by its neighbors is the main distinct property of topological interlocking. The kinematic constraint by neighboring blocks can schematically be represented as simultaneous action of faces (or parts thereof) inclined in the opposite directions, which prevents the removal of a block from the assembly, Fig. 7. The opposite inclination of the faces can be provided either by different neighboring blocks as in the case of Platonic solids or by different parts of the same surface as in the case of osteomorphic blocks. In this setting, the force removing the block should overcome both the resistance of the constraint P over the inclined surface and the friction between the contacting surfaces (see [30] for details). Thus the removing force is

(1)F=nP\tan ({\alpha }_{\max }+\phi ),

where α_max is either α as in the case of Platonic solids where it is constant (but is different for different Platonic bodies) or the maximum inclination angle (in the case of osteomorphic blocks), ϕ is the friction angle (for the sake of simplicity we neglect cohesion) and n is the number of surfaces in contact (in Fig. 7, n = 2).

Two things are obvious from Eq. (1). First, even in the absence of friction the geometry still prevents the block form being removed. Secondly, the higher the angle of interlocking the stronger the resistance to the block removal is. There are, however, limitations on the maximum value of the angle of interlocking posed by 1) the feasibility of manufacture and assembly of the blocks and 2) the stress concentrations created by partial sliding over the contacting surfaces. This is a manifestation of complex stress state created by complex contact patterns. The stress state and, in particular, stress concentrations have been found by direct finite element modeling [29]. Here we provide a simple model that explains their origin.

Suppose a delamination of a length 2l occurred over the interface inclined at the interlocking angle α, Fig. 8. Assume that the constraining force produces compressive stress σ_x. Then the delamination zone will effectively act as a shear crack whose faces slide under shear load \tau ={\sigma }_x(\sin \alpha \cos \alpha -\tan \phi {\cos }^{\mathrm{2}}\alpha ) (when the expression in square brackets is positive). This is the shear stress induced on the plane of the interface minus the friction stress (cohesion is neglected). The stress concentration created by the shear crack can, in the 2D approximation, be characterized by the Mode II stress intensity factor (e.g., [31])

(2)K_{\mathrm{I}\mathrm{I}}=\max [{\sigma }_x\sqrt{\mathrm{\pi }l}(\sin \alpha -\cos \alpha \tan \phi )\cos \alpha ,\mathrm{0}].

At an angle of about -70.5° to the direction of crack continuation a concentration of tensile stress is created [31] which can be characterized by the Mode I stress intensity factor K_{\mathrm{I}}=\mathrm{2}{K}_{\mathrm{I}\mathrm{I}}/\sqrt{\mathrm{3}}. This stress concentration can create wing cracks (shown in Fig. 8 by broken lines) that are able to destroy the block. The condition of generating the wing cracks is K_I = K_IC, where K_IC is the fracture toughness of the material of the block.

Figure 9 shows the normalized K_II as a function of both angles involved. It is seen that the stress intensity factor reaches its maximum at the interlocking angle α_μαζ = π/2+ ϕ/4. The maximum value is K_{\mathrm{I}\mathrm{I}\max }={\sigma }_x\sqrt{\mathrm{\pi }l}(\mathrm{1}-\sin \phi )/\cos \phi. Therefore there are values of the interlocking angle, near α_μαζ that should be avoided to ensure the integrity of the individual blocks.

The above analysis concerns the case of planar interface. In the case of curved interfaces (osteomorphic blocks) the two mechanical effects considered are separate. The resistance to block removal is controlled by the highest interlocking angle attainable, while the stress concentrations may be created by other parts of the contact surface with different values of the interlocking angle, for instance the ones closer to α_μαζ. The interplay between these two mechanisms can be used for the optimisation of interlocking shapes.

Resistance to bending

Topological interlocking structures are segmented structures in that there are no connectors to hold the blocks. While interlocking ensures that the blocks cannot be removed from the structure, it does not prevent partial delamination during bending, as exemplified by Fig. 10. The delamination depends upon the constraining force, P.

To investigate the effect of delamination, we consider a simple 1D structure- a beam fragmented or segmented into unconnected elements, Fig. 7. (The analysis of more realistic 2D structures requires further work.) We use the approach suggested in [32] and model the segmented beam as a homogenized one with bending stiffness that depends upon the magnitude of bending moment, thus accounting for the delamination.

We assume, for the sake of simplicity, that the cross section of the interlocking beam is rectangular. Consider first the ideal case when all interfaces are perpendicular to the neutral axis of the beam. We also restrict ourselves to the simplest case of the width of the contact area coinciding with the width of the beam cross-section. Then the delamination zone is also rectangular, Fig. 11. We approximate the contact stress distribution, in the spirit of the elementary beam theory, by a piece-wise linear function whose linear part ends at the beginning of the delamination zone, y = h.

(3)\sigma \left(y\right)=\{\begin{array}{l}0,ify>h;\hfill \\ {\displaystyle \frac{1}{s}}(y-h),if-b\le y\le h,\hfill \end{array}

where 1/s is the stress gradient.

Both the stress gradient and the length of the delamination zone f = b-h are determined from the equations of force and moment equilibrium

(4)-P=t{\displaystyle {\int }_{-b}^h\sigma (y)\mathrm{d}y},M=t{\displaystyle {\int }_{-b}^h\sigma (y)y\mathrm{d}y},

where P is the constraining force and M is the bending moment applied. Solving Eqs. (3), (4) and generalizing for negative moments one obtains

(5)s={\displaystyle \frac{t}{\mathrm{2}P}}{(\mathrm{2}b-f)}^{\mathrm{2}},f=\mathrm{3}\left|{\displaystyle \frac{M}{P}}\right|-b.

The condition f = 2b corresponds to full delamination. Using (3), (5) one can determine the moment of inertia [33]

(6)EI=\{\begin{array}{ll}{\displaystyle \frac{\mathrm{2}}{\mathrm{3}}}Et{b}^{\mathrm{3}},\hfill & iff=\mathrm{0};\hfill \\ {\displaystyle \frac{\mathrm{9}}{\mathrm{4}}}Et{(b-\left|{\displaystyle \frac{M}{P}}\right|)}^{\mathrm{3}},\hfill & if\mathrm{0}\le f\le \mathrm{2}b;\hfill \\ \mathrm{0},\hfill & iff\ge \mathrm{2}b,\hfill \end{array}

where t is the beam thickness, Fig. 11.

The above analysis is formally derived for the case of the interfaces between the blocks normal to the neutral axis of the beam. If the interface is inclined we consider its projection onto the plane normal to the neutral axis. We note that the shear stresses acting on the normal plane do not contribute to the bending moment. Therefore only normal stress distribution should be considered which brings us back to Eqs. (3)-(6) where σ_y(y) stands for the normal stress on that plane, rather than the contact stress, y is the coordinate axis along the normal plane and f is the length of the projection of the delamination zone.

It is seen that the behavior of fragmented beam can be described by an equivalent monolithic Euler-Bernoulli beam with bending stiffness IE depending upon the bending moment. Deflection, u, of such a beam is described by the following system of nonlinear differential equations:

(7)\{\begin{array}{l}{\displaystyle \frac{{\mathrm{d}}^{\mathrm{2}}}{\mathrm{d}{x}^{\mathrm{2}}}}\left(EI(M){\displaystyle \frac{{\mathrm{d}}^{\mathrm{2}}u}{\mathrm{d}{x}^{\mathrm{2}}}}\right)=w(x);\hfill \\ M=-EI(M){\displaystyle \frac{{\mathrm{d}}^{\mathrm{2}}u}{\mathrm{d}{x}^{\mathrm{2}}}},\hfill \end{array}

where u(x) is deflection, x is the position along the span, w(x) is the distributed load applied to the beam and EI(M) is given by Eq. (6).

The set of Eq. (7) is greatly simplified when the load is only given in terms of bending moment. Then Eq. (7) reduces to

(8)M=-EI(M){\displaystyle \frac{{\mathrm{d}}^{\mathrm{2}}u}{\mathrm{d}{x}^{\mathrm{2}}}}.

As an example, consider a simply supported beam of length L = 1000 mm and cross section width of b = 100mm loaded by a point force applied at the center, Fig. 12(a). The assumed Young’s modulus E = 30GPa and the forces are counted per unit depth (t) of the beam. The constraining force is P = 30000 N/mm. Figure 12(b) shows the dependence of deflection at the center of the beam on the force for two cases: the monolithic (conventional) beam and the segmented beam. It is seen that when the force is low such that the delamination is not yet initiated, the curves for the monolithic and fragmented beams coincide. The force of about 2000 N/mm triggers delamination and subsequently the deflection of the segmented beam starts exceeding that of the monolithic one. The delamination zone is localized at the middle of the beam, Fig. 12(a), where the high bending moments act. (Since the beam is simply supported, the bending moment at the beam ends is zero.)

The maximum bending moment in this example is M_max = FL/2, where L is the beam length. On the other hand, according to the second equation in Eq. (5) the delamination length depends upon the ratio M/P. In the case of loading by point force F, the delamination length depends upon the dimensionless ratio F/P. Since the delamination controls the bending stiffness, the behavior of this segmented beam is governed by the ratio F/P. Thus, an increase in the load can be compensated for by the corresponding increase in the constraining force.

The model presented is (nonlinear) elastic, as it does not produce residual deflection. This is at variance with the experiments [9,14,16] with interlocking plate-like assemblies under central concentrated loading (indentation), which showed that all types of block structures tested (assemblies of tetrahedra [9], cubes [16] and osteomorphic blocks [14]) exhibit both residual deflection and considerable hysteresis. The energy loss associated with the hysteresis can be attributed to either friction between the contacting surfaces or to local non-elastic deformation caused by indenting the face of one block with a vertex of another in the process of block rotation. These observations suggest that a considerable energy damping can be expected when the interlocking structure is subjected to vibration. Recent studies have also shown [34] that assemblies of osteomorphic blocks possess enhanced sound absorption capacity.

Discussion

The structural design based on the principle of topological interlocking enables construction of mortarless structures that do not need special pins or connectors to ensure the integrity of the structure. In addition, owing to their engineered shape, the osteomorphic blocks allow considerable flexibility in the type of structures built. Furthermore, the osteomorphic blocks are self-adjusting, thus making it easy to assemble a structure. Buildings or other civil engineering structures built from topologically interlocked blocks in general, and osteomorphic blocks in particular, are also readily demountable. Practically full recycling of the blocks and re-assembly of the structure is thus possible.

The very nature of topological interlocking dictates the particular properties of the interlocking structures. First and foremost, these structures require peripheral constraint to maintain structural integrity. The constraint can be either external, provided by a specially erected frame or by features of the site topography, or internal, using pre-stressed cables/tendons or simply the self-weight of the structure.

The constraint creates a force resisting the removal of a block. This force depends upon the interlocking angle (or the maximum interlocking angle in the case of curved contact surfaces as in osteomorphic blocks). The choice of interlocking angle is dictated by the magnitude of the resisting force requited and is restricted by the requirements of ease of block manufacturing and assembling. Another restriction on the possible choices of the interlocking angle comes from the fact that the angle influences the magnitude of stress concentrations produced by partial sliding of blocks on the contact surfaces. (This sliding can be frictionless if the sliding parts coincide with the gaps between the bocks left due to manufacturing imperfections.) These stress concentrations produce tensile stresses, which can damage or even split a block. Therefore the load-bearing capacity of interlocking structure is smaller than that of a monolithic structure unless its material is specially reinforced (like fiber-reinforced concrete) to increase the tensile strength and fracture toughness.

The resistance to the removal of a block and the strength of the block material are usually considered as the main factors that ensure structural integrity of an assembly of interlocked blocks. The loading of structure walls can also cause their bending. Traditionally, the effect of bending is considered in terms of the bending failures created. This is justified for such a composite as brick and mortar, but in segmented structures bending does not immediately lead to failure, as was assumed in modeling of the stability of dry stone walls [35]. However, in segmented structures, bending is a process initially producing delamination without failure and reduction in bending stiffness. The reduced bending stiffness leads to deflections considerably larger than the ones characteristic of monolithic structures with the same elastic moduli. At the same time, these large deflections are accompanied by considerable energy dissipation, which probably explains the amazing longevity of historic mortarless structures despite high seismicity of the regions where some of them are located.

These large deflections are controlled not only by the applied load but also by the magnitude of the constraining force. Furthermore, the deflections can be measured and the results of the measurements can be used to monitor the extent of delamination and therefore the health of the structure.

A very important factor- the constraining force (or stress) - now becomes a new design parameter, which can considerably affect the mechanical behavior of the structure. Furthermore, this is a controllable parameter, which can be varied during service of the structure. (This can be done for instance by changing the tensioning stress or by adding surcharge to increase the self-weight.) This opens a possibility to adjust the structural response, for instance by increasing or decreasing the degree of delamination and thus raising or reducing the flexibility of the structure. This will eventually lead to designing smart structures capable of adjusting their behavior to respond to particular external actions or the environmental conditions.

Conclusions

Structures whose design is based on the principle of topological interlocking, especially those constructed from osteomorphic blocks, are easy to assemble (and disassemble). They are flexible, resistant to macroscopic fractures and tolerant to missing blocks. The overall structural integrity of these structures depends on the constraining force provided by a peripheral constraint. The peripheral constraint can be applied in many different ways: by an external frame, features of site topography or internal pre-stressed cables/tendons. An alternative to that is the use or self-weight and, if necessary, artificial surcharge.

Individual blocks in interlocking structures are kept in place by the kinematic constraint from the neighboring blocks due to the specifics of the block shape and the pattern of the assembly. The force resisting removal of a block is proportional to the constraining force externally imposed by the peripheral constraint. The proportionality factor depends upon the maximum value of the interlocking angles that determine the inclination of the parts of contact surfaces to the interlocking plane. On the other hand, partial sliding on the contact surfaces can create stress concentrations in individual blocks and cause their failure. Thus an optimisation procedure needs to be devised to determine the interlocking shape that minimises the stress concentrations while maintaining the required resistance to the block removal.

The structures based on topological interlocking of blocks not connected in any other way respond to bending loading by partial delamination at block interfaces. The onset of delamination does not signify failure: there is a considerable leeway with respect to further increase of loading, which results only in further delamination and increase of deflection.

Both delamination and the resistance to the block removal are governed by the magnitude of the constraining force, which is a new controlling design parameter. Furthermore, by making the constraining force variable one can design smart structures that can adjust their properties in response to environmental challenges or stimuli.

References

[1] Harris H G, Oh K, Hamid A A. Development of new interlocking and mortarless block masonry units for efficient building systems, In: Proceedings of the 6th Canadian Masonry Symposium. Saskatoon, Canada, June 15-17, 1992

[2] Anand K B, Ramamurthy K. Development and performance evaluation of interlocking-block masonry. Journal of Architectural Engineering, 2000, 6(2): 45-51

[3] Gilroy D, Goffi E D. Modular Interlocking Brick System in Wide Use at BHP. AISE Steel Technology, Jan, 2001

[4] Weinhuber K. Building with Interlocking Blocks. German Appropriate Technology Exchange, 1995,

[5] Ramamurthy K, Kunhanandan E. Accelerated masonry construction: review and future prospects. Progress in Structural Engineering and Materials, 2004, 6(1): 1-9

[6] Heyman J. The Stone Skeleton. Structural Engineering of Masonry Architecture. Cambridge: Cambridge University Press, 1997

[7] Brooks A, Adcock S. Dry Stone Walling. 2nd ed. Doncaster UK: BTCV, 1999

[8] Psycharis I N, Papastamatiou D Y, Alexandris A P. Parametric investigation of the stability of classical columns under harmonic and earthquake excitations. Earthquake Engineering & Structural Dynamics, 2000, 29(8): 1093-1109

[9] Dyskin A V, Estrin Y, Kanel-Belov A J, et al. A new concept in design of materials and structures: Assemblies of interlocked tetrahedron-shaped elements. Scripta Materialia, 2001, 44(12): 2689-2694

[10] Dyskin A V, Estrin Y, Kanel-Belov A J, et al. Toughening by fragmentation- How topology helps. Advanced Engineering Materials, 2001, 3(11): 885-888

[11] Khor C, Dyskin A V, Pasternak E, et al. Integrity and fracture of plate-like assemblies of topologically interlocked elements. In: Dyskin A V, Hu X Z, Sahouryeh E, eds. Structural Integrity and Fracture. Swets & Zeitlinger, Lisse, 2002, 449-456

[12] Dyskin A V, Estrin Y, Kanel-Belov A J, et al. Topological interlocking of platonic solids: A way to new materials and structures. Philosophical Magazine Letters, 2003, 83(3): 197-203

[13] Dyskin A V, Estrin Y, Kanel-Belov A J, et al. A new principle in design of composite materials: reinforcement by interlocked elements. Composites Science and Technology, 2003, 63(3-4): 483-491

[14] Dyskin A V, Estrin Y, Pasternak E, et al. Fracture resistant structures based on topological interlocking with non-planar contacts. Advanced Engineering Materials, 2003, 5(3): 116-119

[15] Estrin Y, Dyskin A V, Pasternak E, et al. Topological interlocking of protective tiles for Space Shuttle. Philosophical Magazine Letters, 2003, 83(6): 351-355

[16] Estrin Y, Dyskin A V, Pasternak E, et al. Negative stiffness of a layer with topologically interlocked elements. Scripta Materialia, 2004, 50(2): 291-294

[17] Dyskin A V, Estrin Y, Kanel-Belov A J, et al. Interlocking properties of buckyballs. Physics Letters. [Part A], 2003, 319(3-4): 373-378

[18] Dyskin A V, Estrin Y, Pasternak E, et al. The principle of topological interlocking in extraterrestrial construction. Acta Astronautica, 2005, 57(1): 10-21

[19] Molotnikov A, Estrin Y, Dyskin A V, et al. Percolation mechanism of failure of a planar assembly of interlocked osteomorphic elements. Engineering Fracture Mechanics, 2007, 74(8): 1222-1232

[20] Schaare S, Dyskin A V, Estrin Y, et al. Point loading of assemblies of interlocked cube-shaped elements. International Journal of Engineering Science, 2008, 46(12): 1228-1238

[21] Kanel-Belov A J, Dyskin A V, Estrin Y, et al. Interlocking of convex polyhedra: towards a geometric theory of fragmented solids. Moscow Mathematical Journal, 2010, arXiv:0812.5089v1

[22] Estrin Y, Dyskin A V, Pasternak E. Topological interlocking as a materials design concept. Materials Science and Engineering C, 2011, 31(6): 1189-1194

[23] Goodman R E, Shi G H. Block Theory and its Application to Rock Engineering. Englewood NJ: Prentice-Hall, 1985

[24] Glickman M. The G-block system of vertically interlocking paving, In: Proceedings of the Second International Conference on Concrete Block Paving. Delft, April 10-12, 1984, 345-348

[25] Robson D A. Deutsches Patent DE-AS 25 54 516, 1978

[26] Autruffe A, Pelloux F, Brugger C, et al. Indentation Behaviour of Interlocked Structures Made of Ice: Influence of the Friction Coefficient. Advanced Engineering Materials, 2007, 9(8): 664-666

[27] Dyskin A V, Caballero A. Orthogonal crack approaching an interface. Engineering Fracture Mechanics, 2009, 76(16): 2476-2485

[28] Shackel B. Design and Construction of Interlocking Pavements. London and New York: Elsevier Applied Science, 1990

[29] Dyskin A V, Yong D, Pasternak E, et al. Stresses in topologically interlocking structures: two scale approach. In: Denier J, Finn M D, Mattner T, eds. ICTAM 2008, XXII International Congress of Theoretical and Applied Mechanics, Adelaide, <month>August</month> <day>24-29</day> 2008, CD-ROM Proceedings ISBN 978-0-9805142-1-6, 2008, 10134

[30] Goodman R E. Introduction to Rock Mechanics. 2nd ed. John Wiley & Sons, 1989

[31] Cherepanov G P. Mechanics of brittle fracture. New York: McGraw Hill, 1979

[32] Barnett R L, Hermann P C. Studies in prestressed and segmented brittle structures, IIT Research Institute, Chicago, 1966

[33] Backshall D. Bending Stiffness of Interlocking Structures. Honours Dissertation, UWA. 2009

[34] Carlesso M, Molotnikov A, Krause T, et al. Enhancement of sound absorption properties using topologically interlocked elements. Scripta Materialia, 2012, 66(7): 483-486

[35] Cooper M R. Deflection and failure modes in dry-stone retaining walls. Ground Engineering, 1986, 19(8): 28-33

RIGHTS & PERMISSIONS

Higher Education Press and Springer-Verlag Berlin Heidelberg