Variational Inequalities and their Applications in the Predictive Modeling of Heterogeneous Media

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The research objective of this project is to create a fundamental understanding and methods for predicting the properties of heterogeneous media including precipitated solids, multiphase multifunctional composites and self-assembled crystals, and to exploit such knowledge for new models for heterogeneous media and optimal design of composite materials. Composites are broadly used in daily life ranging from plastic bags and concrete (used as a construction material) to the Boeing 787 Dreamliner whose fuselage is made of carbon composites. The challenge facing researchers today consists of extending this success to smart/multifunctional composites. These are composites that serve several functions and can be excited by and respond to multiple external stimuli. The properties of multifunctional composites are predominantly determined by the microstructure of the constituent phases. The present research systematically addresses how microstructures influence the properties of multifunctional composites and what are the optimal microstructures for a desired property, e.g., the magnetoelectric coupling coefficient. These predictive modeling and optimal design problems are addressed by an inverse method: microstructures are "a priori" designed to find the quantitative relation between microstructures and material properties. Also, a numerical toolset will be developed to compute optimal microstructures for various applications. The bottleneck of many technologies, e.g., batteries and turbine engines, hinges on the development of new multifunctional materials. The results of this research can be used to develop new multifunctional composites and new manufacturing processes for these technologies, which have the potential to mitigate some aspects of the current energy crisis. Educationally, an interactive visualization demonstrations project will be launched to promote students' interest and responsibility to learn engineering.