Distributions of elastic moduli in mechanically percolating composites

Power-law percolation models contain very little mechanics other than the theoretical or simulated value of a percolation threshold, the volume fraction where a connected microstructure forms. For mechanical percolation these theoretical values do not correspond well to experimental results and so the models are commonly used empirically; results are correlative rather than predictive. In recent work, the effective elastic properties of a model polymer nanocomposite were approximated using a computational micromechanics model within a Monte Carlo framework. Significantly, the statistical averages resulting from these simulations displayed distinct percolation-like behavior. Of equal interest is the distribution of properties that resulted from the randomly simulated microstructures. This strongly suggests that mechanical percolation in nanocomposites is the result of a combination of microstructural mechanisms. Analysis aimed at determining which microstructure produces what response is a challenging task if microstructure is the random variable. In this work, the effective composite properties are considered as the random variable; probability distribution functions (PDFs) of the properties at discrete volume fractions are developed using the Principle of Maximum Informational Entropy. The evolution of these PDFs with increasing volume fraction helps visualize and track the significant property changes that result from microstructural randomness.