THE CONCEPT OF TOPOLOGICAL INTERLOCKING IN ENGINEERING

1
THE CONCEPT OF TOPOLOGICAL INTERLOCKING IN
ENGINEERING
AV Dyskin1, Y Estrin2 , E Pasternak1, HC Khor1,
AJ Kanel-Belov3

• 1Department of Civil and Resource Engineering,
  The University of Western Australia, Australia
• 2Institut für Werkstoffkunde und
  Werkstofftechnik, Technische Universität
  Clausthal, Germany
• 3University of Bremen, Germany

2
Assembly of topologically interlocked tetrahedron
elements
This block prevents the reference block from
moving upwards
Reference block
Glickman (1984) pavements
These blocks prevent the reference block from
moving downwards
3
Plan

• Two classes of interlocking shapes
• Osteomorphic blocks
• Assemblies of tetrahedra and their derivatives
• Deformation of assembly of tetrahedra.
  Concentrated load test
• Failure. Tolerance to missing blocks
• Further topologically interlocking polyhedra
• Deformation of assembly of cubes
• Topological interlocking of Buckyballs
• Conclusions

4
Two classes of interlocking shapes

• Interlocking of identical convex elements
• Interlocking by non-planar surfaces (Osteomorphic
  blocks)

z
y
x
5
Assembly of osteomorphic blocks
6
Fracturing of the assembly
7
Testing of a solid plate
It was loaded and unloaded 4 times, then loaded
to failure...
8
Fracture of the plate
9
Tolerance to missing blocks
10
Properties of assemblies of osteomorphic blocks

• High fracture toughness
• High tolerance to missing blocks
• Low bending rigidity
• Self-adjusting
• No porosity
• Versatile applications

11
Assemblies of tetrahedra and their derivatives
Square-based interlocking
12
Circle-based assembly
Middle plane section
Smooth surface of the elements, very high
porosity and permeability of the assembly
Top section
13
Deformation of assembly of tetrahedra.
Concentrated load test
Load Instron machine, cylindrical indentor 1 cm
in diameter
1 cm
A commercial Al-Mg-Si alloy
14
Test results. Assembly
D cylindrical bending stiffness
D 0.34 kNm
D 0.31 kNm
For reference solid plate of the same thickness
as the assembled layer
D 2.33 kNm
15
Test results. Reference solid plate
Reference plate, simply supported edges. The same
thickness as the assembled layer
D 2.33 kNm
D cylindrical bending stiffness
16
Mechanism of low bending stiffness
a
Contact area
Cylindrical bending stiffness
Dsolid plate/D8
First unloading, Dsolid plate/D7.5 Second
unloading, Dsolid plate/D6.9
Test
17
Shape after unloading
18
Residual deflection
Blocks themselves show no plastic deformation
The assembly is still holding The
elements curve in and lock the indentor
19
Profile after unloading
Depth (mm)
20
Failure. Tolerance to missing blocks
Assemblies with missing blocks retain their
integrity.
21
Two possible mechanisms of failure
Accumulation of independently failed blocks (eg,
due to impacts)
Long distance propagation of a crack originated
in a failed block?
22
Long distance crack propagation
Crack retardation
Crack propagation from a fractured block towards
an interface. The concentration of tensile stress
acting in the direction of crack propagation
creates an interface crack
Delamination eventually arrests the propagation
of the main crack
Weak adhesion of the blocks isolates a failed
block and preserves the integrity of the assembly
23
Tolerance to missing blocks
24
Influence of missing blocks
25
Properties of assemblies of tetrahedral blocks

• High fracture toughness
• Tolerance to missing blocks
• Low bending rigidity
• High porosity
• Assemblies are permeable
• Template is needed for assembling

26
Hexagonal based interlocking
Dodecahedron
Cube
Octahedron
27
Sufficient condition of interlocking
28
Assemblies
Blocks 1, 3, 5 prevent downward movement
Blocks 2, 4, 6 prevent upward movement
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Deformation of assembly of cubes
Concentrated load
30
Anomalous unloading curves in assembly of cubes
F N
F
u
Courtesy S. Schaare
u mm
31
Cyclic loadingof PVC cubes
Y
F
u
X
X
Y
Courtesy S. Schaare
32
Residual strain
Localisation of block rotations
Bottom view
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Mechanism of post-peak softening
F
Interlocked block in a section
Elastic sx p lt 0
h0
p
p
sx
F
Critical sx p 0
p
p
sx
F
Post-peak sx p 0, hlth0 Low bending rigidity
h
p
p
34
Possible mechanism of anomalous unloading
F N
u mm
35
Simple model
F, w
k(R,h0)1
p
h?h0
2
wpeak/h00.05
wpeak/h00.3
hh0
R
Total (measured) deflection wwewp, hh0-wp
1.5
hlth0
Elastic deflection
1
wpeak/h00.1
0.5
wpeak/h00.01
Separation condition
sx p 0
wpeak/h00.001
0
0
0.2
0.4
0.6
0.8
1
w/h0
36
Properties of assemblies of cubic blocks

• High fracture toughness
• Low bending rigidity
• Localisation of rotations
• Anomalous unloading response
• High porosity
• Permeable assemblies
• No tolerance to missing blocks
• Template is needed for assembling

37
Decagon based interlocking
Dodecahedron
Buckyball (Truncated icosahedron)
Icosahedron
38
Interlocking of Buckyballs (C60 molecules)
Central section consists of decagons
Weak interlocking
39
Another way of topological interlocking of
Buckyballs
Central section consists of 18-gons
Weak interlocking
40
Conclusions

• Assemblies of interlocked elements form flexible
  layers in which each individual block is held in
  place by neighbouring blocks
• Assemblies posses high fracture toughness weak
  inter-block adhesion leads to arrest of
  propagating cracks
• Some assemblies are tolerant to missing blocks
• Rotational degrees of freedom of the blocks may
  play important role in the deformation response
• The interlocking elements can form skeletons of
  composites with the binder phase ensuring the
  necessary functional properties