Singularly Perturbed and Multiscale Problems

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Proposal: DMS-0405423PI: Giovanni LeoniInstitution: Carnegie-Mellon UniversityTitle: Singularly Perturbed and Multiscale SystemsABSTRACTThe central theme of this proposal is the application of modern mathematics to materials science in order to study the macroscopic behavior of materials at small length scale. The PI proposes the following specific studies: Dislocations. Planned here is a variational formulation of the equilibrium problem for a finite number of dislocations in a plane domain, and the characterization of the energy content of a body with isolated defects in terms of a regular function of the defect configuration, the renormalized energy. Generalized Wulff shapes of second-phase particles and voids in the presence of bulk elasticity. The PI plans to study the influence on corner angles of the interaction between elasticity, surface energy and the corner regularization.Elastic solid-state phase transitions. Planned here is the study of a higher order variational model for the fine scale structure of twinning of two-phase variants of martensite in the presence of an austenite-twinning martensite interface.Micromagnetics. In the study of the microstructure of ferromagnetic materials simplified models, usually derived by neglecting or simplifying various terms in the micromagnetic energy, have been proposed in particular regimes. The PI plans to study the complete micromagnetic energy in order to determine the length scale and the fine geometry of the refinement, as well as domain branching, that is, the separation of scales of the domain widths at the boundary and at the center of the body.Defects in crystals, microstructures, and phase transitions play a key role in determining properties of materials. Thus, understanding their structure is necessary in achieving desired material properties and optimized performance. While electron microscopy and experimental activity have provided a great deal of detailed information on structures, the integration of mathematical modeling, analysis with experimental approaches promises to greatly increase our understanding of phase transformations, ferromagnetic, ferroelectric and martensitic materials, nanomaterials, thin films and self-assembled structures.