We investigate the elastic effective modulus Eeff of two-dimensional checkerboard specimens in which square tiles are randomly assigned to one of two component phases. This is a model system for a wide class of multiphase polycrystalline materials such as granitic rocks and many ceramics. We study how the effective stiffness is affected by different characteristics of the specimen (size relative to the tiles, stiff fraction, and modulus contrast between the phases) and obtain analytical approximations to the probability distribution of Eeff as a function of these parameters. In particular, we examine the role of percolation of the soft and stiff phases, a phenomenon that is important in polycrystalline materials and composites with inclusions. In small specimens, we find that the onset of percolation causes significant discontinuities in the effective modulus, whereas in large specimens, the influence of percolation is smaller and gradual. The analysis is an extension of the elastic homogenization methodology of Dimas et al. (2015, “Random Bulk Properties of Heterogeneous Rectangular Blocks With Lognormal Young's Modulus: Effective Moduli,” ASME J. Appl. Mech., 82(1), p. 011003), which was devised for blocks with lognormal spatial variation of the modulus. Results are validated through Monte Carlo simulation. Compared with lognormal specimens with comparable first two moments, checkerboard plates have more variable effective modulus and are on average less compliant if there is prevalence of stiff tiles and more compliant if there is prevalence of soft tiles. These differences are linked to percolation.

References

References
1.
Beran
,
M. J.
,
1968
,
Statistical Continuum Theories. Monographs in Statistical Physics and Thermodynamics
, Vol.
XV
,
Interscience Publishers
,
New York
, p.
424
.
2.
Kachanov
,
M.
, and
Sevostianov
,
I.
,
2013
,
Effective Properties of Heterogeneous Materials
, Vol.
193
,
Springer
, Dordrecht.
3.
Nemat-Nasser
,
S.
and
Hori
,
M.
,
1993
,
Micromechanics: Overall Properties of Heterogeneous Materials
(North-Holland Series in Applied Mathematics and Mechanics),
North-Holland
,
New York
, p.
687
.
4.
Sanchez-Palencia
,
E.
, and
Zaoui
,
A.
,
1985
,
Homogenization Techniques for Composite Media
(Lectures Delivered at the CISM International Center for Mechanical Sciences),
Springer
,
Berlin
, p.
397
.
5.
Torquato
,
S.
,
2002
,
Random Heterogeneous Materials: Microstructure and Macroscopic Properties
, Vol.
16
,
Springer
, New York.
6.
Dimas
,
L. S.
,
Veneziano
,
D.
,
Giesa
,
T.
, and
Buehler
,
M. J.
,
2015
, “
Random Bulk Properties of Heterogeneous Rectangular Blocks With Lognormal Young's Modulus: Effective Moduli
,”
ASME J. Appl. Mech.
,
82
(
1
), p.
011003
.
7.
Balland
,
M.
,
Desprat
,
N.
,
Icard
,
D.
,
Fereol
,
S.
,
Asnacios
,
A.
,
Browaeys
,
J.
,
Henon
,
S.
, and
Gallet
,
F.
,
2006
, “
Power Laws in Microrheology Experiments on Living Cells: Comparative Analysis and Modeling
,”
Phys. Rev. E
,
74
(
2
), p.
021911
.
8.
Cheung
,
M. S.
, and
Li
,
W. C.
,
2003
, “
Probabilistic Fatigue and Fracture Analyses of Steel Bridges
,”
Struct. Saf.
,
25
(
3
), pp.
245
262
.
9.
Geyskens
,
P.
,
Der Kiureghian
,
A.
, and
Monteiro
,
P.
,
1998
, “
Bayesian Prediction of Elastic Modulus of Concrete
,”
ASCE J. Struct. Eng.
,
124
(
1
), pp.
89
95
.
10.
Hiratsuka
,
S.
,
Mizutani
,
Y.
,
Tsuchiya
,
M.
,
Kawahara
,
K.
,
Tokumoto
,
H.
, and
Okajima
,
T.
,
2009
, “
The Number Distribution of Complex Shear Modulus of Single Cells Measured by Atomic Force Microscopy
,”
Ultramicroscopy
,
109
(
8
), pp.
937
941
.
11.
Ulm
,
F. J.
,
Vandamme
,
M.
,
Bobko
,
C.
, and
Ortega
,
J. A.
,
2007
, “
Statistical Indentation Techniques for Hydrated Nanocomposites: Concrete, Bone, and Shale
,”
J. Am. Ceram. Soc.
,
90
(
9
), pp.
2677
2692
.
12.
Gupta
,
H. S.
,
Stachewicz
,
U.
,
Wagermaier
,
W.
,
Roschger
,
P.
,
Wagner
,
H. D.
, and
Fratzl
,
P.
,
2006
, “
Mechanical Modulation at the Lamellar Level in Osteonal Bone
,”
J. Mater. Res.
,
21
(
8
), pp.
1913
1921
.
13.
Younis
,
S.
,
Kauffmann
,
Y.
,
Bloch
,
L.
, and
Zolotoyabko
,
E.
,
2012
, “
Inhomogeneity of Nacre Lamellae on the Nanometer Length Scale
,”
Cryst. Growth Des.
,
12
(
9
), pp.
4574
4579
.
14.
Nachtrab
,
S.
,
Kapfer
,
S. C.
,
Arns
,
C. H.
,
Madadi
,
M.
,
Mecke
,
K.
, and
Schroder-Turk
,
G. E.
,
2011
, “
Morphology and Linear-Elastic Moduli of Random Network Solids
,”
Adv. Mater.
,
23
(
22–23
), pp.
2633
2637
.
15.
Soulie
,
R.
,
Merillou
,
E.
,
Romain
,
O.
,
Djamchid
,
T.
, and
Ghazanfarpour
,
R.
,
2007
, “
Modeling and Rendering of Heterogeneous Granular Materials: Granite Application
,”
Comput. Graph Forum
,
26
(
1
), pp.
66
79
.
16.
Fan
,
Z. G.
,
Wu
,
Y. G.
,
Zhao
,
X. H.
, and
Lu
,
Y. Z.
,
2004
, “
Simulation of Polycrystalline Structure With Voronoi Diagram in Laguerre Geometry Based on Random Closed Packing of Spheres
,”
Comput. Mater. Sci.
,
29
(
3
), pp.
301
308
.
17.
Kumar
,
S.
, and
Kurtz
,
S. K.
,
1994
, “
Simulation of Material Microstructure Using a 3D Voronoi Tessellation—Calculation of Effective Thermal-Expansion Coefficient of Polycrystalline Materials
,”
Acta Metall. Mater.
,
42
(
12
), pp.
3917
3927
.
18.
Keller
,
J. B.
,
1987
, “
Effective Conductivity of Periodic Composites Composed of Two Very Unequal Conductors
,”
J. Math. Phys.
,
28
(
10
), pp.
2516
2520
.
19.
Kirkpatrick
,
S.
,
1973
, “
Percolation and Conduction
,”
Rev. Mod. Phys.
,
45
(
4
), pp.
574
588
.
20.
Landauer
,
R.
,
1978
, “
Electrical Conductivity in Inhomogeneous Media
,” Electrical Transport and Optical Properties of Inhomogeneous Media,
AIP Conf. Proc.
,
40
, pp.
2
45
..
21.
Sheng
,
P.
,
1980
, “
Theory for the Dielectric Function of Granular Composite Media
,”
Phys. Rev. Lett.
,
45
(
1
), pp.
60
63
.
22.
Sheng
,
P.
, and
Kohn
,
R. V.
,
1982
, “
Geometric Effects in Continuous-Media Percolation
,”
Phys. Rev. B
,
26
(
3
), pp.
1331
1335
.
23.
Berlyand
,
L.
, and
Golden
,
K.
,
1994
, “
Exact Result for the Effective Conductivity of a Continuum Percolation Model
,”
Phys. Rev. B
,
50
(
4
), pp.
2114
2117
.
24.
Chen
,
Y.
, and
Schuh
,
C. A.
,
2009
, “
Effective Transport Properties of Random Composites: Continuum Calculations Versus Mapping to a Network
,”
Phys. Rev. E
,
80
(
4
), p.
040103
.
25.
Malarz
,
K.
, and
Galam
,
S.
,
2005
, “
Square-Lattice Site Percolation at Increasing Ranges of Neighbor Bonds
,”
Phys. Rev. E
,
71
(
1
), p.
016125
.
26.
Veneziano
,
D.
,
2003
,
Computational Fluid and Solid Mechanics
,
Elsevier
, Cambridge, MA.
27.
Berggren
,
S. A.
,
Lukkassen
,
D.
,
Meidell
,
A.
, and
Simula
,
L.
,
2001
, “
A New Method for Numerical Solution of Checkerboard Fields
,”
J. Appl. Math.
,
1
(
4
), pp.
157
173
.
28.
Helsing
,
J.
,
2011
, “
The Effective Conductivity of Arrays of Squares: Large Random Unit Cells and Extreme Contrast Ratios
,”
J. Comput. Phys.
,
230
(
20
), pp.
7533
7547
.
29.
Milton
,
G. W.
,
2002
, “
The Theory of Composites
,”
Cambridge Monographs on Applied and Computational Mathematics
, Vol.
XXVIII
,
Cambridge University Press
,
Cambridge
, p.
719
.
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