Abstract

Architectured materials contain highly controlled structures and morphological features at length scales intermediate between the microscale and the size of the component. In dense architectured materials, stiff building blocks of well-defined size and shape are periodically arranged and bonded by weak but deformable interfaces. The interplay between the architecture of the materials and the interfaces between the blocks can be tailored to control the propagation of cracks while maintaining high stiffness. Interestingly, natural materials such as seashells, bones, or teeth make extensive use of this strategy. While their architecture can serve as inspiration for the design of new synthetic materials, a systematic exploration of architecture-property relationships in architectured materials is still lacking. In this study, we used the discrete element method (DEM) to explore the fracture mechanics of several hundreds of 2D tessellations composed of rigid “tiles” bonded by weaker interfaces. We explored crack propagation and fracture toughness in Voronoi-based tessellations (to represent intergranular cracking in polycrystalline materials), tessellations based on regular polygons, and tessellations based on brick-and-mortar. We identified several toughening mechanisms including crack deflection, crack tortuosity, crack pinning, and process zone toughening. These models show that periodic architectures can achieve higher toughness when compared with random microstructures, the toughest architectures are also the most anisotropic, and tessellations based on brick and mortar are the toughest. These findings are size independent and can serve as initial guidelines in the development of new architectured materials for toughness.

References

References
1.
Ashby
,
M. F.
,
2005
, “
Hybrids to Fill Holes in Material Property Space
,”
Philos. Mag.
,
85
(
26–27
), pp.
3235
3257
. 10.1080/14786430500079892
2.
Evans
,
A. G.
,
Hutchinson
,
J. W.
, and
Ashby
,
M. F.
,
1998
, “
Multifunctionality of Cellular Metal Systems
,”
Prog. Mater. Sci.
,
43
(
3
), pp.
171
221
. 10.1016/S0079-6425(98)00004-8
3.
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
(
12
), pp.
2689
2694
. 10.1016/S1359-6462(01)00968-X
4.
Siegmund
,
T.
,
Barthelat
,
F.
,
Cipra
,
R.
,
Habtour
,
E. M.
, and
Riddick
,
J. C.
,
2016
, “
Manufacture and Mechanics of Topologically Interlocked Material Assemblies
,”
ASME Appl. Mech. Rev.
,
68
(
4
), p.
040803
. 10.1115/1.4033967
5.
Barthelat
,
F.
,
2015
, “
Architectured Materials in Engineering and Biology: Fabrication, Structure, Mechanics and Performance
,”
Int. Mater. Rev.
,
60
(
8
), pp.
413
430
. 10.1179/1743280415Y.0000000008
6.
Ritchie
,
R. O.
,
2011
, “
The Conflicts Between Strength and Toughness
,”
Nat. Mater.
,
10
(
11
), pp.
817
822
. 10.1038/nmat3115
7.
Dyskin
,
A. V.
,
Estrin
,
Y.
,
Kanel-Belov
,
A. J.
, and
Pasternak
,
E.
,
2001
, “
Toughening by Fragmentation—How Topology Helps
,”
Adv. Eng. Mater.
,
3
(
11
), pp.
885
888
. 10.1002/1527-2648(200111)3:11<885::AID-ADEM885>3.0.CO;2-P
8.
Mirkhalaf
,
M.
,
Tanguay
,
J.
, and
Barthelat
,
F.
,
2016
, “
Carving 3D Architectures Within Glass: Exploring New Strategies to Transform the Mechanics and Performance of Materials
,”
Extreme Mech. Lett.
,
7
, pp.
104
113
. 10.1016/j.eml.2016.02.016
9.
Mirkhalaf
,
M.
,
Zhou
,
T.
, and
Barthelat
,
F.
,
2018
, “
Simultaneous Improvements of Strength and Toughness in Topologically Interlocked Ceramics
,”
Proc. Natl. Acad. Sci. U.S.A.
,
115
(
37
), pp.
9128
9133
. 10.1073/pnas.1807272115
10.
Wadley
,
H. N. G.
,
O’Masta
,
M. R.
,
Dharmasena
,
K. P.
,
Compton
,
B. G.
,
Gamble
,
E. A.
, and
Zok
,
F. W.
,
2013
, “
Effect of Core Topology on Projectile Penetration in Hybrid Aluminum/Alumina Sandwich Structures
,”
Int. J. Impact Eng.
,
62
, pp.
99
113
. 10.1016/j.ijimpeng.2013.05.008
11.
Barthelat
,
F.
,
Tang
,
H.
,
Zavattieri
,
P. D.
,
Li
,
C.-M.
, and
Espinosa
,
H. D.
,
2007
, “
On the Mechanics of Mother-of-Pearl: A key Feature in the Material Hierarchical Structure
,”
J. Mech. Phys. Solids
,
55
(
2
), pp.
306
337
. 10.1016/j.jmps.2006.07.007
12.
Weiner
,
S.
,
Traub
,
W.
, and
Wagner
,
H. D.
,
1999
, “
Lamellar Bone: Structure–Function Relations
,”
J. Struct. Biol.
,
126
(
3
), pp.
241
255
. 10.1006/jsbi.1999.4107
13.
Seidel
,
R.
,
Lyons
,
K.
,
Blumer
,
M.
,
Zaslansky
,
P.
,
Fratzl
,
P.
,
Weaver
,
J. C.
, and
Dean
,
M. N.
,
2016
, “
Ultrastructural and Developmental Features of the Tessellated Endoskeleton of Elasmobranchs (Sharks and Rays)
,”
J. Anat.
,
229
(
5
), pp.
681
702
. 10.1111/joa.12508
14.
Chen
,
I. H.
,
Yang
,
W.
, and
Meyers
,
M. A.
,
2015
, “
Leatherback Sea Turtle Shell: A Tough and Flexible Biological Design
,”
Acta Biomater.
,
28
, pp.
2
12
. 10.1016/j.actbio.2015.09.023
15.
Wegst
,
U. G. K.
,
Bai
,
H.
,
Saiz
,
E.
,
Tomsia
,
A. P.
, and
Ritchie
,
R. O.
,
2015
, “
Bioinspired Structural Materials
,”
Nat. Mater.
,
14
(
1
), pp.
23
36
. 10.1038/nmat4089
16.
Barthelat
,
F.
,
Yin
,
Z.
, and
Buehler
,
M. J.
,
2016
, “
Structure and Mechanics of Interfaces in Biological Materials
,”
Nat. Rev. Mater.
,
1
(
4
), p.
16007
. 10.1038/natrevmats.2016.7
17.
Dunlop
,
J. W. C.
,
Weinkamer
,
R.
, and
Fratzl
,
P.
,
2011
, “
Artful Interfaces Within Biological Materials
,”
Mater. Today
,
14
(
3
), pp.
70
78
. 10.1016/S1369-7021(11)70056-6
18.
Bajaj
,
D.
, and
Arola
,
D. D.
,
2009
, “
On the R-Curve Behavior of Human Tooth Enamel
,”
Biomaterials
,
30
(
23–24
), pp.
4037
4046
. 10.1016/j.biomaterials.2009.04.017
19.
Kamat
,
S.
,
Su
,
X.
,
Ballarini
,
R.
, and
Heuer
,
A. H.
,
2000
, “
Structural Basis for the Fracture Toughness of the Shell of the Conch Strombus Gigas
,”
Nature
,
405
(
6790
), pp.
1036
1040
. 10.1038/35016535
20.
Koester
,
K. J.
,
Ager
J. W.
, III, and
Ritchie
,
R. O.
,
2008
, “
The True Toughness of Human Cortical Bone Measured with Realistically Short Cracks
,”
Nat. Mater.
,
7
(
8
), pp.
672
677
. 10.1038/nmat2221
21.
Smith
,
B. L.
,
Schäffer
,
T. E.
,
Viani
,
M.
,
Thompson
,
J. B.
,
Frederick
,
N. A.
,
Kindt
,
J.
,
Belcher
,
A.
,
Stucky
,
G. D.
,
Morse
,
D. E.
, and
Hansma
,
P. K.
,
1999
, “
Molecular Mechanistic Origin of the Toughness of Natural Adhesives, Fibres and Composites
,”
Nature
,
399
(
6738
), pp.
761
763
. 10.1038/21607
22.
Fratzl
,
P.
, and
Weinkamer
,
R.
,
2007
, “
Nature’s Hierarchical Materials
,”
Prog. Mater. Sci.
,
52
(
8
), pp.
1263
1334
. 10.1016/j.pmatsci.2007.06.001
23.
Sen
,
D.
, and
Buehler
,
M.
,
2011
, “
Structural Hierarchies Define Toughness and Defect-Tolerance Despite Simple and Mechanically Inferior Brittle Building Blocks
,”
Sci. Rep.
,
1
(
35
), pp.
1
9
.
24.
Barthelat
,
F.
, and
Mirkhalaf
,
M.
,
2013
, “
The Quest for Stiff, Strong and Tough Hybrid Materials: An Exhaustive Exploration
,”
J. R. Soc. Interface
,
10
(
89
), p.
20130711
. 10.1098/rsif.2013.0711
25.
Gu
,
G. X.
,
Wettermark
,
S.
, and
Buehler
,
M. J.
,
2017
, “
Algorithm-driven Design of Fracture Resistant Composite Materials Realized Through Additive Manufacturing
,”
Addit. Manuf.
,
17
, pp.
47
54
. 10.1016/j.addma.2017.07.002
26.
Gu
,
G. X.
, and
Buehler
,
M. J.
,
2018
, “
Tunable Mechanical Properties Through Texture Control of Polycrystalline Additively Manufactured Materials Using Adjoint-Based Gradient Optimization
,”
Acta Mech.
,
229
(
10
), pp.
4033
4044
. 10.1007/s00707-018-2208-1
27.
Gu
,
G. X.
,
Chen
,
C.-T.
, and
Buehler
,
M. J.
,
2018
, “
De Novo Composite Design Based on Machine Learning Algorithm
,”
Extreme Mech. Lett.
,
18
, pp.
19
28
. 10.1016/j.eml.2017.10.001
28.
Gu
,
G. X.
,
Chen
,
C.-T.
,
Richmond
,
D. J.
, and
Buehler
,
M. J.
,
2018
, “
Bioinspired Hierarchical Composite Design Using Machine Learning: Simulation, Additive Manufacturing, and Experiment
,”
Mater. Horiz.
,
5
(
5
), pp.
939
945
. 10.1039/C8MH00653A
29.
Abid
,
N.
,
Pro
,
J. W.
, and
Barthelat
,
F.
,
2018
, “
Fracture Mechanics of Nacre-Like Materials Using Discrete-Element Models: Effects of Microstructure, Interfaces and Randomness
,”
J. Mech. Phys. Solids
,
124
, pp.
350
365
.
30.
Abid
,
N.
,
Mirkhalaf
,
M.
, and
Barthelat
,
F.
,
2018
, “
Discrete-element Modeling of Nacre-Like Materials: Effects of Random Microstructures on Strain Localization and Mechanical Performance
,”
J. Mech. Phys. Solids
,
112
, pp.
385
402
. 10.1016/j.jmps.2017.11.003
31.
Pro
,
J. W.
,
Kwei Lim
,
R.
,
Petzold
,
L. R.
,
Utz
,
M.
, and
Begley
,
M. R.
,
2015
, “
GPU-Based Simulations of Fracture in Idealized Brick and Mortar Composites
,”
J. Mech. Phys. Solids
,
80
, pp.
68
85
. 10.1016/j.jmps.2015.03.011
32.
Rezakhani
,
R.
, and
Cusatis
,
G.
,
2016
, “
Asymptotic Expansion Homogenization of Discrete Fine-Scale Models With Rotational Degrees of Freedom for the Simulation of Quasi-Brittle Materials
,”
J. Mech. Phys. Solids
,
88
, pp.
320
345
. 10.1016/j.jmps.2016.01.001
33.
Cusatis
,
G.
, and
Schauffert
,
E.
,
2010
, “
Discontinuous Cell Method (DCM) for Cohesive Fracture Propagation
,”
Proceedings of the 7th International Conference on Fracture Mechanics of Concrete and Concrete Structures (FraMCos 7)
,
Korea Concrete Institute, Seoul
,
May 23–28
.
34.
Cusatis
,
G.
,
Pelessone
,
D.
, and
Mencarelli
,
A.
,
2011
, “
Lattice Discrete Particle Model (LDPM) for Failure Behavior of Concrete. I: Theory
,”
Cement Concrete Comp.
,
33
(
9
), pp.
881
890
. 10.1016/j.cemconcomp.2011.02.011
35.
Melosh
,
R. J.
,
1963
, “
Basis for Derivation of Matrices for the Direct Stiffness Method
,”
AIAA J.
,
1
(
7
), pp.
1631
1637
. 10.2514/3.1869
36.
Mathews
,
J. H.
, and
Fink
,
K. D.
,
2004
,
Numerical Methods Using MATLAB
, Vol. 4,
Pearson London
,
UK
.
37.
Press
,
W. H.
,
Teukolsky
,
S. A.
,
Vetterling
,
W. T.
, and
Flannery
,
B. P.
,
2007
,
Numerical Recipes 3rd Edition: The Art of Scientific Computing
,
Cambridge University Press
,
Cambridge, England/London/New York
, p.
1256
.
38.
Tada
,
H.
,
Paris
,
P. C.
, and
Irwin
,
G. R.
,
1973
,
The Stress Analysis of Cracks
,
Handbook, Del Research Corporation
,
Sandusky
.
39.
Nguyen
,
V.-D.
,
Béchet
,
E.
,
Geuzaine
,
C.
, and
Noels
,
L.
,
2012
, “
Imposing Periodic Boundary Condition on Arbitrary Meshes by Polynomial Interpolation
,”
Comput. Mater. Sci.
,
55
, pp.
390
406
. 10.1016/j.commatsci.2011.10.017
40.
Rice
,
J. R.
,
1968
, “
A Path Independent Integral and the Approximate Analysis of Strain Concentration by Notches and Cracks
,”
J. Appl. Mech.
35
(
2
), pp.
379
386
. 10.1115/1.3601206
41.
Kolednik
,
O.
,
Schöngrundner
,
R.
, and
Fischer
,
F.
,
2014
, “
A new View on J-Integrals in Elastic–Plastic Materials
,”
Int. J. Fract.
,
187
(
1
), pp.
77
107
. 10.1007/s10704-013-9920-6
42.
Simha
,
N.
,
Fischer
,
F. D.
,
Shan
,
G. X.
,
Chen
,
C. R.
, and
Kolednik
,
O.
,
2008
, “
J-integral and Crack Driving Force in Elastic–Plastic Materials
,”
J. Mech. Phys. Solids
,
56
(
9
), pp.
2876
2895
. 10.1016/j.jmps.2008.04.003
43.
Anderson
,
T. L.
,
2017
,
Fracture Mechanics: Fundamentals and Applications
,
CRC Press
,
Boca Raton, FL
.
44.
Gao
,
Y.
, and
Bower
,
A.
,
2004
, “
A Simple Technique for Avoiding Convergence Problems in Finite Element Simulations of Crack Nucleation and Growth on Cohesive Interfaces
,”
Modell. Simul. Mater. Sci. Eng.
,
12
(
3
), pp.
453
. 10.1088/0965-0393/12/3/007
45.
Zhang
,
P.
,
Balint
,
D.
, and
Lin
,
J.
,
2011
, “
An Integrated Scheme for Crystal Plasticity Analysis: Virtual Grain Structure Generation
,”
Comput. Mater. Sci.
,
50
(
10
), pp.
2854
2864
. 10.1016/j.commatsci.2011.04.041
46.
Burger
,
G.
,
Koken
,
E.
,
Wilkinson
,
D. S.
, and
Embury
,
J. D.
,
2013
, “
The Influence of Spatial Distributions on Metallurgical Processes
,”
Advances in Phase Transitions: Proceedings of the International Symposium Held at McMaster University Ontario
,
Canada
,
Oct. 22–23, 1987
,
Elsevier
,
New York/Amsterdam
.
47.
Du
,
Q.
,
Emelianenko
,
M.
, and
Ju
,
L.
,
2006
, “
Convergence of the Lloyd Algorithm for Computing Centroidal Voronoi Tessellations
,”
SIAM J. Numer. Anal.
,
44
(
1
), pp.
102
119
. 10.1137/040617364
48.
Alsayednoor
,
J.
, and
Harrison
,
P.
,
2016
, “
Evaluating the Performance of Microstructure Generation Algorithms for 2-D Foam-Like Representative Volume Elements
,”
Mech. Mater.
,
98
, pp.
44
58
. 10.1016/j.mechmat.2016.04.001
49.
Grünbaum
,
B.
, and
Shephard
,
G. C.
,
1987
,
Tilings and Patterns
,
Freeman
,
San Francisco
.
50.
Keaveny
,
T. M.
,
Morgan
,
E. F.
,
Niebur
,
G. L.
, and
Yeh
,
O. C.
,
2001
, “
Biomechanics of Trabecular Bone
,”
Annu. Rev. Biomed. Eng.
,
3
(
1
), pp.
307
333
. 10.1146/annurev.bioeng.3.1.307
51.
Srivastava
,
A.
,
Ponson
,
L.
,
Osovski
,
S.
,
Bouchaud
,
E.
,
Tvergaard
,
V.
, and
Needleman
,
A.
,
2014
, “
Effect of Inclusion Density on Ductile Fracture Toughness and Roughness
,”
J. Mech. Phys. Solids
,
63
, pp.
62
79
. 10.1016/j.jmps.2013.10.003
52.
E1820-18ae1, A.
, “
Standard Test Method for Measurement of Fracture Toughness
,”
ASTM International
.
53.
Blackman
,
B.
,
Dear
,
J. P.
,
Kinloch
,
A. J.
,
Macgillivray
,
H.
,
Wang
,
Y.
,
Williams
,
J. G.
, and
Yayla
,
P.
,
1995
, “
The Failure of Fibre Composites and Adhesively Bonded Fibre Composites Under High Rates of Test
,”
J. Mater. Sci.
,
30
(
23
), pp.
5885
5900
. 10.1007/BF01151502
54.
De Morais
,
A.
, and
Pereira
,
A.
,
2007
, “
Application of the Effective Crack Method to Mode I and Mode II Interlaminar Fracture of Carbon/Epoxy Unidirectional Laminates
,”
Composites, Part A
,
38
(
3
), pp.
785
794
. 10.1016/j.compositesa.2006.09.001
55.
Heide-Jørgensen
,
S.
,
de Freitas
,
S. T.
, and
Budzik
,
M. K.
,
2018
, “
On the Fracture Behaviour of CFRP Bonded Joints Under Mode I Loading: Effect of Supporting Carrier and Interface Contamination
,”
Compos. Sci. Technol.
,
160
, pp.
97
110
. 10.1016/j.compscitech.2018.03.024
56.
Hossain
,
M.
,
Hsueh
,
C.-J.
,
Bourdin
,
B.
, and
Bhattacharya
,
K.
,
2014
, “
Effective Toughness of Heterogeneous Media
,”
J. Mech. Phys. Solids
,
71
, pp.
15
32
. 10.1016/j.jmps.2014.06.002
57.
Davidson
,
P.
, and
Waas
,
A. M.
,
2012
, “
Non-smooth Mode I Fracture of Fibre-Reinforced Composites: an Experimental, Numerical and Analytical Study
,”
Phil. Trans. R. Soc. A
,
370
(
1965
), pp.
1942
1965
. 10.1098/rsta.2011.0381
58.
Heide-Jørgensen
,
S.
, and
Budzik
,
M. K.
,
2018
, “
Effects of Bondline Discontinuity During Growth of Interface Cracks Including Stability and Kinetic Considerations
,”
J. Mech. Phys. Solids
,
117
, pp.
1
21
. 10.1016/j.jmps.2018.04.002
59.
Naleway
,
S. E.
,
Porter
,
M. M.
,
McKittrick
,
J.
, and
Meyers
,
M. A.
,
2015
, “
Structural Design Elements in Biological Materials: Application to Bioinspiration
,”
Adv. Mater.
,
27
(
37
), pp.
5455
5476
. 10.1002/adma.201502403
60.
Deville
,
S.
,
Saiz
,
E.
,
Nalla
,
R. K.
, and
Tomsia
,
A. P.
,
2006
, “
Freezing as a Path to Build Complex Composites
,”
Science
,
311
(
5760
), pp.
515
518
. 10.1126/science.1120937
61.
Plummer
,
H. C.
,
1950
,
Brick and Tile Engineering: Handbook of Design
,
Structural Clay Products Institute
,
Washington, DC
.
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