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A generalized differential effective medium theory. (English) Zbl 0579.73114
Summary: A generalization of the differential effective medium approximation (DEM) is discussed. The new scheme is applied to the estimation of the effective permittivity of a two phase dielectric composite. Ordinary DEM corresponds to a realizable microgeometry in which the composite is built up incrementally through a process of homogenization, with one phase always in dilute suspension and the other phase associated with the percolating backbone. The generalization of DEM assumes a third phase which acts as a backbone. The other two phases are progressively added to the backbone such that each addition is in an effectively homogeneous medium. A canonical ordinary differential equation is derived which describes the change in material properties as a function of the volume concentration \(\phi\) of the added phases in the composite. As \(\phi\) \(\to 1\), the effective medium approximation (EMA) is obtained. For \(\phi <1\), the result depends upon the backbone and the mixture path that is followed. The approach to EMA for \(\phi\) \(\cong 1\) is analysed and a generalization of Archie’s law for conductor-insulator composites is described. The conductivity mimics EMA above the percolation threshold and DEM as the conducting phase vanishes.

MSC:
74F15 Electromagnetic effects in solid mechanics
74E05 Inhomogeneity in solid mechanics
74A40 Random materials and composite materials
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