Percolation and homogenization theories for heterogeneous materials

Abstract

Most materials produced by Nature and by human beings are heterogeneous. They contain domains of different states, structures, compositions, or material phases. How these different domains are distributed in space, or in other words, how they connect to one another, determines their macroscopic properties to a large degree, making the simple rule-of-mixtures ineffective in most cases. This thesis studies the macroscopic effective diffusion, diffusional creep, and elastic properties of heterogeneous grain boundary networks and composite solids, both theoretically and numerically, and explores the microstructure-property correlations focusing on the effects of microstructural connectivity (topology). We have found that the effects of connectivity can be effectively captured by a percolation threshold, a case-specific volume fraction at which the macroscopic effective property undergoes a critical transition, and a set of critical scaling exponents, which also reflect the universality class that the property belongs to. Using these percolation quantities together with the generalized effective medium theory, we are able to directly predict the effective diffusivity and effective diffusional creep viscosity of heterogeneous grain boundary networks to a fairly accurate degree. Diffusion in composite solids exhibits different percolation threshold and scaling behaviors due to interconnectivity at both edges and corners. Continuum elasticity suffers from this complexity as well, in addition to the complicating factor that each phase is always characterized by several independent elastic constants. These issues are each addressed in detail. In addition to studying all the above properties for a random distribution of grain boundaries or phases, we have also studied the effects of correlations in spatial distributions.

(cont.) This topic is especially important in materials science, because virtually no materials exhibit random phase distributions. We have examined the percolation of effective properties for correlated microstructures spanning between the random distribution and the perfectly periodic distribution. An important result of this work is new understanding about what correlations may be considered small, or inconsequential, to the percolation scaling behavior, and which are large or long-range, and lead to a loss of universality. Finally, a rigorous, and easy-to-use, analytical homogenization method is developed for periodic composite materials.