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Probabilistic Engineering Mechanics, Volume 30, October 2012, Pages 1-8
Bethany S. Fralick, Edward P. Gatzke, Sarah C. Baxter
Department of Mechanical Engineering, University of South Carolina, SC 29208, United States
Department of Chemical Engineering, University of South Carolina, SC 29208, United States
Abstract
One mechanism that is expected to play a large role in the enhanced properties of nanocomposites is the formation of percolated or connected microstructures. Traditional percolation models, well developed for modeling electrical conductivity, are largely empirical and have little foundation in mechanics. Analytic micromechanics models, without the ability to describe random microstructural arrangement do not capture effects associated with the evolution of a percolated microstructure. In this work, a unit cell micromechanics model is used to predict the effective composite properties of simulated random microstructures in particulate reinforced composites. Scale effects, which are present in nanocomposites, are introduced into the model through the inclusion of an interfacial region linked to the size of the reinforcing phase. By tracking and observing the variability in the predicted effective properties due to random microstructures, the onset and evolution of mechanical percolation and related microstructural events can be tracked.
Additional Information
Sharp increases in mechanical properties have been experimentally observed in nanocomposite materials at uncharacteristically low volume fractions.
Because this behavior strongly resembles the sigmoidal curves associated with percolation theory and percolation of electrical conductivity in composites, it has been dubbed ‘mechanical percolation’. As a result, this behavior has been almost exclusively modeled using power-law functions based on the mathematical formulation from percolation theory, and effective media models from electrical composites. These models are dominated by a percolation threshold or critical volume fraction, which marks the probabilistic onset of the formation of a connected, or percolated, microstructure.
There are several challenges to using these models to describe mechanical properties. First, the theoretical values of the percolation threshold and percolation exponent are not universal for continuum percolation and don’t match well to experimental results, which limits the model use to the empirical. Second, there is very little mechanics, other than connectivity, in most of these models; this ignores other mechanical based microstructural mechanisms. Finally, extensions to the power-law formulation in the literature still require prior knowledge of a valid percolation threshold.
In this work we have reversed the modeling paradigm, instead of predicting mechanical properties from the percolation threshold, a percolation threshold is predicted from the mechanical properties of simulated random microstructures. A computational micromechanics model is used to simulate random microstructures and develop approximations of effective elastic properties.
The results show a clear percolation effect for a two-phase composite, particle and matrix. The response is significantly different than predictions of classic mean-field micromechanics models, but occurs at a relatively high volume fraction. Stacked particles produce the maximum moduli, but a distribution of moduli develops even when these microstructures are not present. For nanocomposites the effect of an interface zone between particle and matrix, with intermediate properties is expected to have a significant effect on composite properties. When a three-phase composite, particle, matrix and interface, is modeled, percolation-like effects appear at relatively low volume fractions, similar to those observed experimentally. Maximum values result when particles and interface form connected microstructures, pseudo-percolation. Both sets of results support hypotheses that the mechanisms behind `mechanical percolation’ are more diverse and potentially complex than simple connectivity.
