Effective properties of heterogeneous materials with applications in electromagnetics
Abstract: Composite materials, i.e, mixtures of two or more materials, are commonly used in industry, because they often have outstanding properties in comparison with the original materials. In many cases the inhomogeneities are on a very fine scale, which makes it difficult to perform a full numerical simulation of the material. On the fine scale we have rapid oscillations in the material parameters, but we are usually interested in the behavior at a much larger scale. At the large scale the composite reacts in the same way as a homogeneous material, with some effective material properties. In this thesis, models for the effective electromagnetic properties are analyzed. Mathematically, it is the study of Maxwell's equations with rapidly oscillating coefficients. The geometry on the fine scale is unknown in the study of many man-made materials and for almost all materials in nature. When, for example, only the permittivity of the components is known but nothing about the geometry, we have bounds on the effective permittivity of the composite, that is, the permittivity of the composite cannot exceed the permittivity of the components. The effective properties of heterogeneous materials depend strongly on the microstructure. This dependence can be quantified in terms of structural parameters, such as the volume fraction and the anisotropy of the material. We discuss the possibility of bounding the structural parameters from measurements of bulk properties of a two-component composite. Moreover, we show that this method can be used in practice, not only to bound the structural parameters but the method also implies restrictions on the possible values of the components in the composite. The problem of bounding the structural parameters from known values of an effective property is called inverse homogenization. Information from measurements of one effective property can be used to improve bounds on a related property. These bounds are called cross-property bounds or coupled bounds. We use the bounds on the structural parameters to derive cross-property bounds for anisotropic materials. When the microstructure is periodic and completely known it is in principle possible to exactly determine the effective properties of the composite. Two different methods for determination of the effective properties are compared numerically and an extension from the static limit of one of the methods is given. Using Bloch waves, the extension is from the static limit (zero wave vector) to an arbitrary wave vector in the first Brillouin zone. A nonzero wave vector is necessary when the microstructure cannot be considered infinitely small compared to the wavelength, for example in the study of optically active materials
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