Homogenization relations for elastic properties based on two -point statistical functions

In this research, the homogenization relations for elastic properties of isotropic and anisotropic materials including composites and polycrystalline materials are studied by applying two-point statistical mechanics theory. The validity of the results is investigated by direct comparison with experimental results.;In today's technology, where advanced processing methods can provide materials with a variety of morphologies and features in different scales, a methodology to link properties to microstructure is necessary to develop a framework for material design. The link between structure of materials in any length scale (from nano to macro) and their properties whether they are mechanical, electrical, magnetic, or optical is critical in every engineering discipline. For this purpose, this research is focused on the homogenization relationships based on two-point statistical information to correlate the microstructure of the materials to their mechanical properties. Statistical distribution functions are commonly used for the representation of microstructures and also for homogenization of materials properties. The use of two-point statistics allows the materials designer to include the morphology and distribution in addition to the properties of the individual phases and components. Statistical mechanics modeling not only enables us to correlate the morphology of the microstructures to properties, it can also predict the microstructures from the properties. The latter issue which is called inverse structure-property problem has received a lot of attention in materials community in recent years.;Microstructure design based on statistical mechanics facilitates and optimizes choosing the microstructures of materials for specific design with desired properties. Therefore studying the statistical mechanics theory in different length scale becomes very important.;In this research, the main focus was to study the effect of one-point and two-point statistics on homogenization relationship for elastic properties of materials. Applying the homogenization relations to the microstructure of simulated isotropic and anisotropic composites, the mathematical representation of two-point probability functions was modified in anisotropic composites and the contribution of one-point and two-point statistics in the calculation of elastic properties was studied. Then, this methodology was applied to two samples of Al-SIC composites which were fabricated by extrusion (PSR: 2:1 and PSR: 8:1). Finally, the technique was extended to completely random and textured polycrystalline materials and the effect of cold rolling on the annealing texture of near-alpha Titanium alloy was presented.;It was shown analytically and numerically that the two-point statistics measurement does not contribute to the calculation of elastic properties in isotropic composites and random polycrystalline materials; however, its contribution is significant in anisotropic composites and textured polycrystalline materials (70% more than the contribution of one-point statistics). Furthermore, the results show that the two-point statistics can represent the effect of clustering in properties in two anisotropic samples of Al-SiC composite. Although the volume fraction of the two samples was the same, two-point statistics was able to capture the morphology of both microstructures and predict the differences in their elastic modulus and shear modulus. In addition, it was shown that the contribution of two-point statistics in calculation of elastic properties of textured polycrystalline is much smaller than its contribution for anisotropic composite materials. All the final results were compared to several micromechanics models. Comparing the computational results to experimental results shows that this methodology is a good tool for structure-property relationships, and can lead to the design new materials with optimized properties as a fundamental backbone to microstructure design.