Morphological and multiscale modeling of stochastic complex materials

Modeling and design of complex materials across multiple length scales are becoming one of the most active engineering research topics. This study develops innovative models for multiscale materials from perspectives of morphologies (PART I) and mechanics (PART II), respectively.; In PART I, an empirical iterative method of translation model is first developed to simulate random media based on the first two orders of statistics, where the limitations of the model are further discussed. For higher-order statistical simulation, a short-range correlation (SRC) model is introduced in the framework of Markov/Gibbs random field theory to characterize and simulate random media. The topological space (X, X ) on a set X of morphological configurations is endowed with a ℓ2 metric induced by the distance between two short-range windowed correlation functions. The Metropolis spin-flip algorithm is applied to build a robust simulator of combinatorial optimization for multiphase materials. Through the SRC model, several crucial conceptual ambiguities are clarified, and higher-order statistical simulation becomes computationally feasible. Based on the numerical simulation, further conjectures are made concerning some fundamental morphological questions, particularly for future investigation of physical behavior.; In PART II, computational solvers for stochastic partial differential equations are applied to multiscale modeling techniques. Incorporation of stochastic processes into multiscale modeling with accurate assessment of uncertainty propagation through nonlinear systems across scales would be a challenging but indispensable task for many applications. By assuming periodicity of heterogeneity, numerical methods based on multiscale homogenization theory have been well established. As far as real problems characterized with random processes are concerned, stochastic homogenization has mostly stayed at the stage of pure mathematical formulations without giving a numerical recipe. To provide a computational stochastic homogenization procedure, a concept of stochastic representative volume element (SRVE) is proposed to provide a numerical approach incorporating uncertainty propagation through nonlinear systems across scales. By replacing the volume average with an ensemble average and applying the Galerkin formulation in probability space, an efficient numerical scheme is implemented using an iterative algorithm that takes advantage of fast Fourier transform. With the proposed model, both global effective properties and local probabilistic behavior of random media can be conveniently evaluated.