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Research Papers

# The Effective Modulus of Random Checkerboard Plates

[+] Author and Article Information
Leon S. Dimas

Laboratory for Atomistic and Molecular Mechanics (LAMM),
Department of Civil and Environmental Engineering,
Massachusetts Institute of Technology,
77 Massachusetts Avenue,
Cambridge, MA 02139;
Department of Civil and Environmental Engineering,
Massachusetts Institute of Technology,
77 Massachusetts Avenue,
Cambridge, MA 02139

Daniele Veneziano

Department of Civil and Environmental Engineering,
Massachusetts Institute of Technology,
77 Massachusetts Avenue,
Cambridge, MA 02139

Markus J. Buehler

Laboratory for Atomistic and Molecular Mechanics (LAMM),
Department of Civil and Environmental Engineering,
Massachusetts Institute of Technology,
77 Massachusetts Avenue,
Cambridge, MA 02139;
Department of Civil and Environmental Engineering,
Massachusetts Institute of Technology,
77 Massachusetts Avenue,
Cambridge, MA 02139
e-mail: mbuehler@MIT.EDU

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received August 25, 2015; final manuscript received October 2, 2015; published online November 5, 2015. Editor: Yonggang Huang.

J. Appl. Mech 83(1), 011007 (Nov 05, 2015) (12 pages) Paper No: JAM-15-1450; doi: 10.1115/1.4031744 History: Received August 25, 2015; Revised October 02, 2015

## Abstract

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.

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## Figures

Fig. 1

Illustration of checkerboard systems and boundary conditions. This is a model system for several polycrystalline systems such as granitic rocks and ceramic materials.

Fig. 2

Examples of ANOVA decomposition of the log-modulus field: (a) {L, p, Δ}={4, 0.5, 1} and (b) {L, p, Δ}={32, 0.25, 1}. F¯ is the spatial average of the log-modulus field, ε1 and ε2 are the marginal fluctuations in the x1- and x2-directions, respectively, ε12 is the second-order fluctuation of F, and F1=F¯+ε1+ε2 is the first-order approximation of the modulus field F.

Fig. 3

Comparison of q12 values for lng=ε12 and lng=F−F¯ when L=4, Δ=4, and the stiff volume fraction f progressively increases from 0 to 1 in 1/16 increments

Fig. 4

Contribution of various ANOVA components to the total effective log-stiffness Feff for (a) L=4 and (b) L=32. In each case Δ = 4. Through numerical simulation, we find the contribution of each term on the right-hand side of Eq. (5) as the stiff volume fraction f progressively increases from 0 to 1 in 1/16 increments. See text for an explanation of the different components.

Fig. 5

Three simulations of Feff(f) for systems with L=4 and Δ=4. The first two columns show the first (NN+NNN)- and NN-percolating clusters. The points along the Feff(f) trajectory at which these percolating clusters form are shown with matching colors. The vertical dashed lines indicate the theoretical percolation thresholds.

Fig. 6

Same as Fig. 5 for L=32. For L=32, the Feff(f) plots are almost deterministic; hence, we display only one simulation.

Fig. 7

(a) Dependence of q12 on fs for L=32 and Δ={1,2,3,4}. (b) Dependence of q12 on f for L={4,8,16,32} and Δ={1,2,3,4} along with regression lines from Eq. (7). (c) Values of c(L) in Eq. (6) for L={4,8,16,32} and values of σq12(L) for L={4,8,16,32}, with least-squares fits.

Fig. 8

Scatter plots of single-value predicted and simulated Feff for L={4,8,16,32}, f={0.1,0.3,0.6,0.9}, and Δ={2,4}

Fig. 9

Comparison of approximate analytical and empirical cumulative distributions of Feff for L={4,8,16,32}, p={0.1,0.3,0.6,0.9}, and Δ={2,4}

Fig. 10

Dependence of Feff on σF2 for large checkerboard and lognormal specimens with mE=1. For checkerboard materials, results are presented for three values of the stiff fraction f. Checkerboard results are shown as solid lines. The dashed lines give hybrid results when the exponent q12 is from checkerboard analysis but the marginal distribution of the modulus is assumed to be lognormal. The dashed middle line is also the log-effective stiffness in the lognormal case. Small panels compare the checkerboard (binary) and normal marginal distributions of F for σF2=0.25 and f={0.1,0.5,0.9}.

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