Topologically interlocked materials (TIMs) are a class of 2D mechanical crystals made by a structured assembly of an array of polyhedral elements. The monolayer assembly can resist transverse forces in the absence of adhesive interaction between the unit elements. The mechanical properties of the system emerge as a combination of deformation of the individual unit elements and their contact interaction. The present study presents scaling laws relating the mechanical stiffness of monolayered TIMs to the system characteristic dimensions. The concept of thrust line analysis is employed to obtain the scaling laws, and model predictions are validated using finite element simulations as virtual experiments. Scaling law powers were found to closely resemble those of classical plate theory despite the distinctly different underlying mechanics and theory of TIM deformation.
Issue Section:
Research Papers
References
1.
Damasceno
, P. F.
, Engel
, M.
, and Glotzer
, S. C.
, 2012
, “Predictive Self-Assembly of Polyhedra Into Complex Structures
,” Science
, 337
, pp. 453
–457
.10.1126/science.12208692.
Clark
, T. D.
, Tien
, J.
, Duffy
, D. C.
, Paul
, K. E.
, and Whitesides
, G. M.
, 2001
, “Self-Assembly of 10μm Sized Objects Into Ordered Three-Dimensional Arrays
,” J. Am. Chem. Soc.
, 123
, pp. 7677
–7682
.10.1021/ja010634l3.
Randhawa
, J. S.
, Gurbani
, S. S.
, Keung
, M. D.
, Demers
, D. P.
, Leahy-Koppa
, M. R.
, and Gracias
, D. H.
, 2010
, “Three-Dimensional Surface Current Loops Terahertz Responsive Microarrays
,” Appl. Phys. Lett.
, 96
, p. 191108
.10.1063/1.34286574.
Nesterenko
, V. F.
, 2001
, Dynamics of Heterogeneous Materials
, Springer
, New York
.5.
Daraio
, C.
, Nesterenko
, V. F.
, Herbold
, E. B.
, and Jin
, S.
, 2005
, “Strongly Nonlinear Waves in a Chain of Teflon Beads
,” Phys. Rev. E
, 72
, p. 016603
.10.1103/PhysRevE.72.0166036.
Merkel
, A.
, Tournat
, V.
, and Gusev
, V.
, 2010
, “Elastic Waves in Noncohesive Frictionless Granular Crystals
,” Ultrasonics
, 50
, pp. 133
–138
.10.1016/j.ultras.2009.09.0327.
Goncu
, F.
, Luding
, S.
, and Bertoldi
, K.
, 2012
, “Exploiting Pattern Transformation to Tune Phononic Band Gaps in a Two-Dimensional Granular Crystal
,” J. Acoust. Soc. Am.
, 131
, pp. EL475
–EL480
.10.1121/1.47183848.
Mueggenburg
, K. E.
, Jaeger
, H. M.
, and Nagel
, S. R.
, 2002
, “Stress Transmission Through Three-Dimensional Ordered Granular Arrays
,” Phys. Rev. E
, 66
, p. 031304
.10.1103/PhysRevE.66.0313049.
Goldenberg
, C.
, and Goldhirsch
, I.
, 2002
, “Force Chains, Microelasticity, and Macroelasticity
,” Phys. Rev. Lett.
, 89
, p. 084302
.10.1103/PhysRevLett.89.08430210.
Ashby
, M. F.
, and Brechet
, Y. J. M.
, 2003
, “Designing Hybrid Materials
,” Acta Mater.
, 51
, pp. 5801
–5821
.10.1016/S1359-6454(03)00441-511.
Mayer
, G.
, 2005
, “Rigid Biological Systems as Models for Synthetic Composites
,” Science
, 310
, pp. 1144
–1147
.10.1126/science.111699412.
Liu
, X.
, Dean
, M. N.
, Summers
, A. P.
, and Earthman
, J. C.
, 2010
, “Composite Model of the Shark's Skeleton in Bending: A Novel Architecture for Biomimetic Design of Functional Compressional Bias
,” Mater. Sci. Eng. C
, 30
, pp. 1077
–1084
.10.1016/j.msec.2010.05.00613.
Zhu
, D.
, Ortega
, C. F.
, Motamedi
, R.
, Szewciw
, L.
, Vernerey
, F.
, and Barthelat
, F.
, 2011
, “Structure and Mechanical Performance of a ‘Modern’ Fish Scale
,” Adv. Eng. Mater.
, 13
, pp. B185
–B194
.10.1002/adem.20100030014.
Levin
, S. M.
, 2002
, “The Tensegrity-Truss as a Model for Spine Mechanics: Biotensegrity
,” J. Mech. Med. Biol.
, 2
, pp. 375
–388
.10.1142/S021951940200047215.
Krauss
, S.
, Monsonego-Ornan
, E.
, Zelzer
, E.
, Fratzl
, P.
, and Shahar
, R.
, 2009
, “Mechanical Function of a Complex Three-Dimensional Suture Joining the Bony Elements in the Shell of the Red-Eared Slider Turtle
,” Adv. Mater.
, 21
, pp. 407
–412
.10.1002/adma.20080125616.
Budd
G. E.
, 2001
, “Why are Arthropods Segmented?
” Evol. Dev.
, 3
, pp. 332
–342
.10.1046/j.1525-142X.2001.01041.x17.
Dyskin
, A. V.
, Estrin
, Y.
, Kanel-Belov
, A. J.
, and Pasternak
, E.
, 2003(a)
, “Topological Interlocking of Platonic Solids: A Way to New Materials and Structures
,” Philos. Mag. Lett.
, 83
, pp. 197
–203
.10.1080/095008303100006522618.
Fleury
, F.
, 2009
, “Evaluation of the Perpendicular Flat Vault Inventor's Intuitions Through Large Scale Instrumented Testing
,” Proceedings of the 3rd International Congress on Construction History
, Cottbus, Germany, May 20–24, Huerta
, S.
, ed., BTU Cottbus
, Cottbus
, Germany, pp. 611
–618
.19.
Brocato
, M.
, and Mondardini
, L.
, 2012
, “A New Type of Stone Dome Based on Abeille's Bond
,” Int. J. Solids Struct.
, 49
, pp. 1786
–1801
.10.1016/j.ijsolstr.2012.03.03620.
Glickman
, M.
, 1984
, “The G-Block System of Vertically Interlocking Paving
,” Proceedings of the 2nd International Conference on Concrete Block Paving
, Delft, Netherlands
, April 10–12, pp. 345
–348
.21.
Dyskin
, A. V.
, Estrin
, Y.
, Kanel-Belov
, A. J.
, and Pasternak
, E.
, 2001
, “A New Concept in Design of Materials and Structures: Assemblies of Interlocked Tetrahedron-Shaped Elements
,” Scr. Mater.
, 44
, pp. 2689
–2694
.10.1016/S1359-6462(01)00968-X22.
Dyskin
, A. V.
, Estrin
, Y.
, Pasternak
, E.
, Khor
, H. C.
, and Kanel-Belov
, A. J.
, 2003(b)
, “Fracture Resistant Structures Based on Topological Interlocking With Non-Planar Contacts
,” Adv. Eng. Mater.
, 5
, pp. 116
–119
.10.1002/adem.20039001623.
Dyskin
, A. V.
, Estrin
, Y.
, Kanel-Belov
, A. J.
, and Pasternak
, E.
, 2003(c)
, “A New Principle in Design of Composite Materials: Reinforcement by Interlocked Elements
,” Comput. Sci. Technol.
, 63
, pp. 483
–491
.10.1016/S0266-3538(02)00228-224.
Khor
, H. C.
, 2008
, “Mechanical and Structural Properties of Interlocking Assemblies
,” Ph.D. thesis, The University of Western Australia, Perth, Australia.25.
Molotnikov
, A.
, Estrin
, Y.
, Dyskin
, A. V.
, Pasternak
, E.
, and Kanel-Belov
, A. J.
, 2007
, “Percolation Mechanism of Failure of a Planar Assembly of Interlocked Osteomorphic Elements
,” Eng. Fract. Mech.
, 74
, pp. 1222
–1232
.10.1016/j.engfracmech.2006.07.01226.
Estrin
, Y.
, Dyskin
, A. V.
, and Pasternak
, E.
, 2011
, “Topological Interlocking as a Material Design Concept
,” Mater. Sci. Eng. C
, 31
, pp. 1189
–1194
.10.1016/j.msec.2010.11.01127.
Schaare
, S.
, Dyskin
, A. V.
, Estrin
, Y.
, Arndt
, S.
, Pasternak
, E.
, and Kanel-Belov
, A.
, 2008
, “Point Loading of Assemblies of Interlocked Cube-Shaped Elements
,” Int. J. Eng. Sci.
, 46
, pp. 1228
–1238
.10.1016/j.ijengsci.2008.06.01228.
Brugger
, C.
, Fivel
, M. C.
, and Brechet
, Y.
, 2009
, “Numerical Simulations of Topologically Interlocked Materials Coupling DEM Methods and FEM Calculations: Comparison With Indentation Experiments
,” MRS Proceedings
, Vol. 1188, 1188-LL05-05, pp. 143–148.10.1557/PROC-1188-LL05-0529.
Mather
, A.
, Cipra
, R. J.
, and Siegmund
, T.
, 2011
, “Structural Integrity During Remanufacture of a Topologically Interlocked Material
,” Int. J. Struct. Integrity
, 3
, pp. 61
–78
.10.1108/1757986121121000930.
Khandelwal
, S.
, Siegmund
, T.
, Cipra
, R. J.
, and Bolton
, J. S.
, 2012
, “Transverse Loading of Cellular Topologically Interlocked Materials
,” Int. J. Solids Struct.
, 49
, pp. 2394
–2403
.10.1016/j.ijsolstr.2012.04.03531.
Timoshenko
, S.
, and Woinowsky-Kreiger
, S.
, 1959
, Theory of Plates and Shells
, 2nd ed., McGraw-Hill Companies
, New York
.32.
Young
, W. C.
, and Budynas
, R. G.
, 2002
, Roark's Formulas for Stress and Strain
, 7th ed., McGraw-Hill Companies
, New York
.33.
Block
, P.
, Ciblac
, T.
, and Ochsendorf
, J.
, 2006
(a), “Real-Time Limit Analysis of Vaulted Masonry Buildings
,” Comput. Struct.
, 84
, pp. 1841
–1852
.10.1016/j.compstruc.2006.08.00234.
Block
, P.
, DeJong
, M.
, and Ochsendorf
, J.
, 2006
(b), “As Hangs the Flexible Line: Equilibrium of Masonry Arches
,” Nexus Network J.
, 8
, pp. 13
–24
.10.1007/s00004-006-0015-935.
Block
, P.
, and Ochsendorf
, J.
, 2007
, “Thrust Network Analysis: A New Methodology for Three-Dimensional Equilibrium
,” J. Int. Assoc. Shell Spatial Struct.
, 48
, pp. 167
–173
.36.
Boothby
, T. E.
, Atamtuerktuer
, H.
, and Erdogmus
, E.
, 2007
, “Manual for the Assessment of Load Bearing Unreinforced Masonry Structures (2007-06)
,” National Center for Preservation Technology and Training (NCPTT), Natchitoches, LA.37.
Casapulla
, C.
, and D'Ayala
, D.
2001
, “Lower-Bound Approach to the Limit Analysis of 3D Vaulted Block Masonry Structures
,” Computer Methods in Structural Masonry
, Vol. 5
, T. G.
Hughes
and G. N.
Pande
, eds., Rome
, Italy
, pp. 177
–183
.38.
Heyman
, J.
, 1966
, “The Stone Skeleton
,” Int. J. Solids Struct.
, 2
, pp. 249
–279
.10.1016/0020-7683(66)90018-739.
Huerta
, S.
, 2001
, “Mechanics of Masonry Vaults: The Equilibrium Approach
,” Proceedings of the 3rd International Seminar of Historical Constructions
, Guimarães, Portugal, November 7–9, P. B.
Lourenço
and P.
Roca
, eds., University of Minho, Guimarães
, Portugal
, pp. 47–70.40.
Livesley
, R. K.
, 1992
, “A Computational Model for the Limit Analysis of Three-Dimensional Masonry Structures
,” Meccanica
, 27
, pp. 161
–172
.10.1007/BF0043004241.
O'Dwyer
, D.
, 1999
, “Funicular Analysis of Masonry Vaults
,” Comput. Struct.
, 73
, pp. 187
–197
.10.1016/S0045-7949(98)00279-X42.
Conway
, J. H.
, and Torquato
, S.
, 2006
, “Packing, Tiling and Covering With Tetrahedra
,” Proc. Natl. Acad. Sci.
, 103
, pp. 10612
–10617
.10.1073/pnas.060138910343.
Ashby
, M. F.
, 2011
, Materials Selection in Mechanical Design
, 4th ed., Butterworth-Heinemann
, Burlington, MA.44.
Bendsoe
, M. P.
, and Sigmund
, O.
, 2003
, Topology Optimization: Theory, Methods, and Applications
, Springer-Verlag
, Berlin
.45.
Hooke
, R.
, 1675
, A Description of Helioscopes, and Some Other Instruments
, by T.R. for John Martyn, Printer to the Royal Society
, London
.46.
Robertson
, R. E.
, Chu
, T. J.
, Gerard
, R. J.
, Kim
, J. H.
, Park
, M.
, Kim
, H. G.
, and Peterson
, R. C.
, 2000
, “Three-Dimensional Fiber Reinforcement Shapes Obtainable From Flat, Bidirectional Fabrics Without Wrinkling or Cutting. Part 1: A Single 4-Sided Pyramid
,” Comp. Part A: App. Sci. & Manuf.
, 31
, pp. 703
–715
.10.1016/S1359-835X(00)00013-047.
Robertson
, R. E.
, Chu
, T. J.
, Gerard
, R. J.
, Kim
, J. H.
, Park
, M.
, Kim
, H. G.
, and Peterson
, R. C.
, 2000
, “Three-Dimensional Fiber Reinforcement Shapes Obtainable From Flat, Bidirectional Fabrics Without Wrinkling or Cutting. Part 2: A Single N-Sided Pyramid, Cone, or Round Box
,” Comp. Part A: App. Sci. & Manuf.
, 31
, pp. 1149
–1165
.10.1016/S1359-835X(00)00120-248.
Randhawa
, J. S.
, Kanu
, L. N.
, Singh
, G.
, and Gracias
, D. H.
, 2010
, “Importance of Surface Patterns for Defect Mitigation in Three-Dimensional Self-Assembly
,” Langmuir
, 26
, pp. 12534
–12539
.10.1021/la101188z49.
Rycenga
, M.
, McLellan
, J. M.
, and Xia
Y.
, 2008
, “Controlling the Assembly of Silver Nanocubes Through Selective Functionalization of Their Faces
,” Adv. Mater.
, 20
, pp. 2416
–2420
.10.1002/adma.200800360Copyright © 2014 by ASME
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