Mesoscale Statistics and Scaling Laws in Planar Elastic Checkerboards

Abstract

Miniaturization and the need for novel materials with unique properties have driven composite materials to the forefront of research in solid mechanics. The response of a composite microstructure is dependent on the properties of individual phases, their distribution, the volume fractions, and the scale of observation. At finite scales the response of a microstructure is realization dependent and statistical in nature. The microstructures under investigation are sampled randomly from an infinite two-phase linear elastic planar checkerboard with a nominal volume fraction of 50%using a binomial distribution. A versatile methodology for investigating the effective response of such microstructures at finite scales is based on the Hill-Mandel Macrohomogenity condition. In this methodology, rigorous bounds are obtained as solutions to stochastic Dirichlet and Neumann boundary value problems from the level of a Statistical Volume Element (SVE) to that of a Representative Volume Element (RVE). Within the framework of planar elasticity, the concept of a scaling function is introduced which unifies the treatment of several microstructures and quantifies the approach to the RVE. It is demonstrated that the scaling function depends on the phase contrast and the mesoscale. Certain exact properties of the scaling function are derived rigorously and its functional form is established using extensive numerical simulations on 163,728 microstructural realizations at varying contrasts, mesoscale and boundary conditions. The statistical nature of the effective response of a microstructure is examined through histograms that illustrate the distributions of the effective stiffness and compliance tensor components and their dependence on mesoscale and contrast in phase properties. Statistical moments (mean, variance, skewness, and kurtosis) are calculated to qualitatively and quantitatively describe the distributions of the tensor components. The Hellinger distance is used to measure the difference between the actual data distribution and a symmetric binomial distribution that is a consequence of the sampling process.