A framework for stochastic mechanics

Interest in and use of multiscale and multiphase materials has necessitated the need for more accurate modeling of the effects of the associated heterogeneous microstructures. Predicting the behavior of these materials is important not only with respect to specific applications but also as the basis for new material design. Towards this end, a review of a combined methodology for the analysis of the bulk and local constitutive behavior of stochastic composites is presented. The first part of the approach is a new stochastic transformation field analysis (STFA) that directly considers the effects of mechanical and transformation field concentration tensors and their associated probability distribution functions on the local and bulk material behavior. The second component of the work utilizes a combined Moving Window based, Generalized Method of Cells (MW/GMC) analysis to sample field data needed to analyze concentration tensors required by the STFA. The GMC micromechanics model was chosen as the basis for the moving window analysis since it provides straightforward access to values of the elements of the concentration tensors for the individual phases in a computationally efficient manner. pabilities of the proposed modeling framework are illustrated through an application to a digital image of a numerically generated microstructure for a two-phase, continuous fiber, composite material. For this simple case, it can be shown analytically that the complete stochastic description for the history-dependent material behavior reduces to a description of the distribution functions for the mechanical concentration tensor. The local variations in the elements of the concentration tensor, due to the random microstructure, are presented. Mean values of these elements are used to predict the bulk material properties, which are compared to established bounds. The effects of the moving window size on the predicted material behavior are considered. This initial analysis considers only the elastic response of the composite.