The Effective Modulus of Random Checkerboard Plates

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.