Description of the material’s internal structure lies at the core of materials science. Recent advances have made it possible to capture the 3-D material structure spanning a multitude of length scales (e.g. X-ray micro-tomography, automated serial sectioning, 3-D atom probe). Advances in material science have led to the development of hundred thousands of distinct engineered and natural materials of interest. A computationally efficient approach for archival, retrieval and real-time exploration of the microstructure datasets is required. Efforts in this direction are hindered by the lack of a rigorous mathematical definition of the internal structure or microstructure of a material. This thesis outlines a rigorous mathematical framework for definition of microstructure as a stochastic process. In this framework the microstructure can be thought of as a set of statistical rules that govern the spatial placement of microstructure features, and observed micrographs are different realizations of the overriding process. This interpretation of microstructure allows for the quantitative comparison of different materials based on structure and more importantly allows for the quantification of the observed variance in samples with the same nominal processing history.

Structure-property correlations are defining concepts in the field of material science, but we still lack to have a well-defined systematic framework for quantitative comparison of microstructures from different material classes. Thus, a novel microstructure quantification framework devised on the lines of framework, published by Niezgoda et. al. (Niezgoda et al., 2013), that facilitates visualization of complex microstructure relationships within and across material classes has been mentioned here. In addition the framework will be used to link microstructure visualizations with properties to develop reduced-order microstructure-property linkages and performance models.