Mathematical Modeling of Dielectric Characteristics of the Metallic Band Inclusion Composite

Among the desirable properties of functional materials used in various electrical and radio physical equipment and devices, dielectric characteristics, including relative permittivity (hereinafter, permittivity) are of importance. The permittivity requirements can be met when a composite with a particular combination of its matrix characteristics and inclusions [1, 2, 3] is used as a functional material. The use of metallic inclusions extends a variation range of dielectric characteristics of the composite, and thereby enhances its application. The composite structure, form of inclusions, and their volume concentration has a significant impact on the permittivity.One of the composite structure embodiments is a dispersion system when in the dispersion medium (in this case | in the composite matrix) a dispersed phase (inclusions) with highly extended interface between them [4] is distributed. There can be various forms of dispersed inclusions. Band is one of the possible forms of inclusion when its dimensions in three orthogonal directions are significantly different among themselves. For such inclusion, a tri-axial ellipsoid can be taken as an acceptable geometric model to describe its form. This model can be used, in particular, to describe the form of nanostructured elements, which recently are considered as inclusions for advanced composites for various purposes [5].With raising volume concentration of metal inclusions in the dielectric matrix composite there is an increasing probability of direct contact between the inclusions resulting in continuous conductive cluster [3, 6]. In this paper, it is assumed that metal band inclusions are covered with a sufficiently thin layer of the electrically insulating material, eliminating the possibility of direct contact and precluding consideration of the so-called percolation effect [2, 7] in the entire interval of the expectedly changing volume concentration of electrically ellipsoidal inclusions. The structural model of the composite these inclusions are replaced by the uniform ellipsoidal inclusions with equivalent anisotropic dielectric characteristics that with the ordered arrangement of the inclusions leads to anisotropy of effective dielectric characteristics of the composite as a whole.There are known various approaches [1, 8, 9, 10] to the mathematical modeling that allow us to build calculated curves to determine dielectric characteristics of the composites having inclusions of different forms. When building such models, the analogy between the formulations and problem solutions of electrostatics and steady thermal conductivity [11, 12, 13, 14] can be used. Variation approaches [15, 16, 17] to estimate effective dielectric properties of the composite allow us to obtain bilateral borders between which there are their true values, and evaluate the maximum possible error occurring in using a particular mathematical model. Such borders can be set on the basis of the dual variation formulation of the problem for a potential field in an inhomogeneous solid [18]. This formulation contains two alternative functionals (minimized and maximized), taking the same extreme values in the true problem solving.DOI: 10.7463/mathm.0515.0815604