Failure of materials is in many cases associated with initiation and subsequent propagation of macroscopic fractures. Consequently, in order to increase the strength, one needs to inhibit either crack initiation or propagation. The principle of topological interlocking provides a unique opportunity to construct materials and structures in which both routes of the strength increase can be realised. Materials and structures built on the basis of this principle consist of many elements which are hold together by the special geometry of their shape, together with an external constrain. The absence of the binder phase between the elements allows the interfaces to arrest macroscopic crack propagation. In addition, with sufficiently small size of the elements an increase in local strength and, possibly, in the stress for crack initiation can be achieved by capitalising on the size effect. Furthermore, the ability of some interlocking structures to tolerate missing elements can serve to prevent the avalanche-type failure initiated by failure of one of the elements. In this paper, experimental results and a theoretical analysis with regard to this possibility are presented

Dyskin, A. V.

Estrin, Y.

Khor, H. C.

Pasternak, E.

University of Queensland eSpace

SIF2004 Structural Integrity and Fracture. http://eprint.uq.edu.au/archive/00000836 
  
Mechanisms Of Fracturing In Structures Built From 
Topologically Interlocked Blocks 
H.C. Khor(1), A.V. Dyskin(1), Y. Estrin(1,2), E. Pasternak(1) 
(1) School of Civil and Resource Engineering, The University of Western Australia, 35 Stirling Highway, 
Crawley WA 6009, Australia 
(2) Institut für Werkstoffkunde und Werkstofftechnik, Technische Universität Clausthal, D-38678 Clausthal-
Zellerfeld, Germany 
 
 
 
ABSTRACT: Failure of materials is in many cases associated with initiation and subsequent propagation of 
macroscopic fractures. Consequently, in order to increase the strength, one needs to inhibit either crack 
initiation or propagation. The principle of topological interlocking provides a unique opportunity to construct 
materials and structures in which both routes of the strength increase can be realised. Materials and structures 
built on the basis of this principle consist of many elements which are hold together by the special geometry 
of their shape, together with an external constrain. The absence of the binder phase between the elements 
allows the interfaces to arrest macroscopic crack propagation. In addition, with sufficiently small size of the 
elements an increase in local strength and, possibly, in the stress for crack initiation can be achieved by 
capitalising on the size effect. Furthermore, the ability of some interlocking structures to tolerate missing 
elements can serve to prevent the avalanche-type failure initiated by failure of one of the elements. In this 
paper, experimental results and a theoretical analysis with regard to this possibility are presented.  
 
1. INTRODUCTION 
Failure of brittle materials and structures is in many cases associated with initiation of a macrocrack 
and its propagation to failure. Correspondingly, an increase in structural stability can be achieved 
either by preventing the formation of macrocracks or by inhibiting their propagation. An efficient 
method of inhibiting macrocrack propagation and thus increasing the macroscopic fracture 
toughness of the material is to introduce internal interfaces with loosely connected opposite faces 
capable of arresting a macrocrack. The role of such interfaces can be played by cracks in a heavily 
cracked body or by internal boundaries in a fragmented body with unbonded fragments, however 
such a body will be either very weak or not in one piece. The principle of topological interlocking 
provides a unique opportunity to construct crack arresting fragmented materials and structures 
which keep their integrity. Materials and structures built according to this principle consist of many 
elements which are held together by their special geometry and an external constraint. A great 
benefit of such design is that the absence of the binder phase between the elements allows the 
interfaces to arrest macroscopic crack propagation. (Note that when a binder phase is introduced to 
join the elements, it offers a medium where the crack can form and propagate.) Furthermore, one 
can capitalise on the size effect by employing elements of sufficiently small dimensions, leading to 
an increased local strength and, possibly, the crack initiation stress [Ashby and Bréchet, 2003]. Both 
recipes for enhancing strength - preventing the formation of the macrocrack or by inhibiting its 
propagation - can thus be implemented using topological interlocking. Furthermore, the ability of 
some interlocking structures to tolerate defects, such as missing elements, can serve to prevent 
avalanche-type failure initiated by failure of an individual element.   
 Topological interlocking is achieved either by special arrangements of convex blocks of simple 
habitus, such as platonic shapes [Dyskin et al., 2003a], or by blocks with specially designed curved 
SIF2004 Structural Integrity and Fracture. http://eprint.uq.edu.au/archive/00000836 
surfaces [Dyskin et al., 2003b] such that if a plate-like assembly is constrained at the periphery, no 
block can be removed from the assembly because its movement is obstructed by the neighbouring 
blocks. An example of the first type of interlocking is given by plate-like assemblies of tetrahedra 
[eg, Glickman, 1984; Dyskin et al., 2001]. The second type of interlocking is exemplified by a set 
of so-called osteomorphic bricks [Robson, 1978; Khor et al., 2002; Dyskin et al., 2003b] Figure 1, 
permitting production of both plate-like and corner-like structures, as well as columns. In the 
osteomorphic block structures, each curved surface of a block obstructs its removal in both 
directions normal to the assembly. 
 
 
(a) 
 
(b) 
Figure 1: Osteomorphic blocks: (a) plate-like structure; (b) full osteomorphic block and two types of half-block 
necessary to furnish the edges. 
 
 Point loading applied at the middle of a plate-like structure made of osteomorphic blocks 
clumped at the edges showed that failure is localised within the blocks on which load is exerted, 
while the assembly as a whole still keeps its integrity [Khor et al., 2002]. This presents a striking 
contrast to the failure mode of a solid reference plate under the same type of loading in which 
fractures transverse the entire plate. The load bearing capacity of the fragmented plate was, 
however, much lower than the one of the solid plate. The present paper analyses the reasons for the 
low load bearing capacity and presents results of a different test where the peak load was similar or 
in one instance greater than the one for a solid plate. 
 
2. MECHANISM OF LOW LOAD BEARING CAPACITY IN LARGE DEFLECTION 
BENDING TESTS 
In bending tests [Khor et al., 2002] the plate-like structure of interlocking blocks is clamped and 
fixed at the edges, and subjected to in-plane compression q from the edges and the displacement-
controlled indentation producing a force F in the middle of the structure. Two main features were 
observed. Firstly, the bearing capacity of the interlocking plate was much lower than the one of the 
reference solid plate of the same thickness made of the same material. Secondly, the deflections 
(indentor displacements) achieved were rather large, close to the thickness of the structure as 
opposite to small deflections of the reference solid plate.  
 Our analysis of the mechanism of low bearing capacity of the interlocking plate will be based 
on the fact that in both cases the fractures were initiated in the indentation area (In the case of 
interlocking structure the fractures were confined within the loaded block.) These fractures are 
obviously caused by the indentor-generated local tensile stresses. The tensile stresses produce 
fractures when their magnitude exceeds the value σt+q, where σt is the tensile strength of the 
material and q is the constraining pressure, both being the same for the interlocking structure and 
SIF2004 Structural Integrity and Fracture. http://eprint.uq.edu.au/archive/00000836 
the reference solid plate. Since the only difference between these two cases is the presence of large 
(comparable with the plate thickness) deflections in the interlocking plate, it is natural to look for 
the mechanism in the non-linear effects related to the large deflections. 
 We consider large deflections of a plate with fixed boundaries under point loading. It is known 
[e.g. Timoshenko and Gere, 1961; Landau and Lifshitz, 1959] that in this case the neutral plane is 
no longer stress-free, but is rather subjected to tensile (membrane) stress, ∆σ, Figure 2, as opposed 
to the cases of cylindrical bending of a plate or bending of a beam. This is a pure geometrical 
feature: cylindrical bending of a plate does not change its perimeter, while bending of a circular 
plate with fixed perimeter under a central load causes the plate diameter to stretch and thus 
produces tensile stresses. When deflection is small, this stress is negligible, however, it becomes 
noticeable when the deflection, ζ(r), is no longer small as compared to the plate thickness.  
 
 
r
R
ζ
q
 
(a) 
F
r
ζ∆σ
q
 
(b) 
Figure 2: Mechanism of membrane stresses in a plate with clamped perimeter: (a) circular plate of radius R under 
confining pressure q before indentation loading; (b) circular plate under indentation force F. Large plate deflection ζ(r) 
induces membrane stresses ∆σ, which reduce the confining pressure and, hence, the resistance to tensile fractures. 
 
 
 In order to estimate this stress, we consider a circular plate of thickness h clamped and fixed at 
the edges. Deflection of such a plate in a polar co-ordinate frame with the origin at the plate centre, 
Figure 2b, in the approximation of small deflections has the form [e.g. Landau and Lifshitz, 1959]: 
  ( ) ( )2
3
2
22
112
,ln
28 νπζ −=⎟⎟⎠
⎞
⎜⎜⎝
⎛ −−= EhD
r
RrrR
D
Fr , (1) 
where E, ν are the (effective) Young’s modulus and Poisson’s ratio of the structure. 
 In the first approximation, the stress ∆σ can be obtained by substituting the small deflection 
approximation (1) into a balance equation of the plate, which in a corresponding Cartesian co-
ordinate frame with x and y axes in the plate plane has the form [e.g. Landau and Lifshitz, 1959] 
  ,,,,0
2
2
2
2
222
2
2
2
2
2
yxxyyxyx
E xyyx ∂∂
∂−=∆∂
∂=∆∂
∂=∆=⎥⎥⎦
⎤
⎢⎢⎣
⎡
⎟⎟⎠
⎞
⎜⎜⎝
⎛
∂∂
∂−∂
∂
∂
∂+∆ χσχσχσζζζχ  (2) 
where χ denotes the Airy stress function.  Transferring (2) to polar co-ordinates, substituting (1) in 
the result and taking into account that due to central symmetry, χ is independent of the polar angle, 
one has 
  ( ) rrrrr
R
r
Rr
D
EF ′′=∆=⎥⎦
⎤⎢⎣
⎡ −+∆ χχπχ
1,0ln1ln
16 22
2
2 . (3) 
 Using the boundary condition, ∆σr=0, and the condition of boundness of stresses at the origin, 
one obtains the circumferential stress component ∆σθ: 
SIF2004 Structural Integrity and Fracture. http://eprint.uq.edu.au/archive/00000836 
  ⎥⎥⎦
⎤
⎢⎢⎣
⎡ −+−⎟⎠
⎞⎜⎝
⎛−=∆ 222
2
22
2
64
13
64
51ln
8
5ln
4
3
64
Rr
r
Rr
r
Rr
D
EF
πσθ . (4) 
 The following condition of fracture generation can be identified near the plate centre (r=0): 
  ( )2
22
6
22
256
1117,)0(,,)0( π
νκκσσσσσ θθ −==∆=+=∆+ Eh
RFkFq FtF . (5) 
Here it is assumed that the tensile stress generated by the indentation, σF, is proportional to the 
force, F, the proportionality constant being denoted by k. From this, the critical failure force, Fcr, is 
given by 
  [ ] ( )6222,121 Ehk qRk qF ttcr +=−++= σκξξξσ . (6) 
 Figure 3 shows the dependence of the normalised failure force, kFcr(σt+q)-1, on  ξ. It is seen 
that for low ξ, which corresponds to large values of Young’s modulus, E, and, hence, low 
deflections, Fcr≈(σt+q) k-1 holds. As E is reduced, the failure force monotonically decreases tending 
to zero. That is why in the interlocking structures, which are characterised by low values of 
Young’s modulus, the force of fracture initiation is considerably lower than in solid plates. 
Fortunately, in contrast with solid plates, in the interlocking structures the fracture initiation does 
not result in complete failure. 
 
0 20 40 60 80
0
0.5
1
ξ
σt+q
kFcr
 
 
Figure 3: Normalised failure force vs. ξ=2κR2(σt+q)/(k2Eh6). 
 
 The above analysis suggests that if deflection is prevented by an external constraint, the force 
of fracture initiation will be similar in both interlocking and solid plates. This is investigated 
experimentally in the following section.  
 
3. COMPRESSION TESTS 
Solid plates and plate-like interlocking structures of rectangular shape, of size 280 x 260 x 20mm 
were loaded by an indentor. These structures rested on a flat surface such that no deflection was 
possible. Both the solid plate and the interlocking blocks were cast from cement paste and cured for 
21 days in a humidity room. Two indentor sizes were used: 10 mm diameter (thin cylinder) and 15 
mm diameter (thick cylinder). The results of the tests are shown in Table 1.  
 
 
SIF2004 Structural Integrity and Fracture. http://eprint.uq.edu.au/archive/00000836 
Table 1. Results of the indentation tests on plate-like structures rested on a flat surface. 
 Peak Strength (kN) 
Test 1 
Peak Strength (kN) 
Test 2 
Peak Strength (kN) 
Average 
Osteomorphic Blocks with Thin 
Cylinder 
6.9 6.7 6.8 
Solid Plate with Thin Cylinder 7.0 - 7.0 
Osteomorphic Blocks with Thick 
Cylinder 
12.3 11.56 11.93 
Solid Plate with Thick Cylinder 9.9 - 9.9 
 
 As evident from Table 1, in these tests the peak loads for the solid plate and the interlocking 
assembly were indeed similar. Furthermore, in the case of the thick cylinder indentor the peak load 
for the interlocking structure was 20% higher that for the solid plate. At the peak load fractures 
were generated, which in the case of the interlocking assembly were contained within the block 
immediately subjected to loading, while other blocks remained intact, Figure 4. By contrast, the 
solid plate was completely fractured, Figure 5. This result displays the superior properties of the 
interlocking assembly where only local fracture at the site of loading was observed, while the 
overall structure still retained its integrity. Furthermore, the interlocking structure could withstand 
multiple fractures before the whole structure ultimately failed.  
 
            
         (a)               (b) 
Figure 4: Assembly of osteomorphic blocks loaded by (a) thin and (b) thick cylindrical indentors. 
            
           
         (a)                (b) 
Figure 5: Solid plate loaded by (a) thin and (b) thick cylindrical indentors. 
 
SIF2004 Structural Integrity and Fracture. http://eprint.uq.edu.au/archive/00000836 
4. CONCLUSION 
In indentation tests the reduced failure load of interlocking structures is associated with their low 
bending rigidity leading to deflections which are comparable with the thickness of the plate. Such 
deflections create appreciable tensile stresses within the plate which counteract the stabilising effect 
of the applied constraining pressure. This suggests that the most efficient use of the interlocking 
structures is in protective coatings and covers when the effect of enhanced fracture toughness and 
tolerance to missing blocks appears to be beneficial, especially when there is a risk of multiple 
damages, for example in a war situation. Essentially, the principle of topological interlocking 
permits manufacturing of fragmented bodies kept in one piece without connectors with their 
tendency to act as stress-concentrators. Then, as soon as the fragmented body is created, one can 
capitalise on two favourable properties, viz. the enhanced fracture toughness and the scale effect, 
when by making the blocks smaller one can increase their strength and consequently the load at 
which fracture initiates.  
 
ACKNOWLEDGEMENTS: Support from the Australian Research Council through Discovery 
Grant DP0210574 (2002-2004) and Linkage International Grant LX0347195 (2002-2004) is 
acknowledged. 
 
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structures: Assemblies of interlocked tetrahedron-shaped elements. Scripta Materialia, 44 (2001) pp. 
2689-2694. 
Dyskin, A.V., Estrin, Y., Kanel-Belov, A.J. and Pasternak, E. Topological interlocking of platonic solids: A 
way to new materials and structures. Phil. Mag. Letters 83(3) (2003a) pp. 197-203. 
Dyskin, A.V., Estrin, Y., Pasternak, E., Khor, H.C. and Kanel-Belov, A.J. Fracture resistant structures based 
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A.V., Hu, X.Z. and Sahouryeh, E., Eds), Swets & Zeitlinger, Lisse (2002) pp. 449-456. 
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Mechanisms Of Fracturing In Structures Built From Topologically Interlocked Blocks

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Mechanisms Of Fracturing In Structures Built From Topologically Interlocked Blocks

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