Mathematical Sciences: Microstructure and its Influence on Composite Material Properties

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DMS-9501025 Milton This proposal aims to investigate microstructures which have novel or extremal elastic behavior, develop new variational principles as a tool for analysis, derive bounds on the complex shear modulus of viscoelastic composites, obtain exact relations between plane elasticity anti-plane elasticity and piezoelectricity, seek a more general link between coupled equations and uncoupled equations, and numerically solve coupled field problems through the use of integral equation methods. Composite and polycrystalline materials are ubiquitous in nature and amongst engineered materials. Their technological importance stretches from applications in earthquake prediction (many rocks are polycrystalline),in the oil industry (rocks containing oil), the aerospace industry (lightweight composites), to understanding sea ice, a composite of brine and ice (important to climate modelling), colloids (a composite of suspended particles in a fluid),to shape memory materials (a composite of different variants of a metallic phase), to hydrophones (composites of piezoelectric materials), to biological applications (bones, shells, tendons are all examples of composite structures). This proposal will enhance our knowledge of composite and polycrystalline materials in ways that will facillitate the development of new materials, and improve procedures for estimating the properties of composites.