Mathematical Modeling of Electrical Conductivity of Dielectric with Dispersed Metallic Inclusions

Composites are increasingly used for application in engineering as structural, thermal protection and functional materials, including dielectrics, because of a wide variety of properties. The relative dielectric constant and the dielectric loss tangent are basic functional characteristics of a composite used as a dielectric. The quantitative level of these characteristics is mainly affected by the properties of the composite matrix and inclusions as well as their shape and volume concentration. Metallic inclusions in a dielectric, which serves as a function of the composite matrix, expand electrical properties of the composite in particular increase its dielectric constant and dielectric loss tangent and thereby greatly expand its application field. Dielectric losses are defined by the imaginary component of the complex value of the relative dielectric constant of the dielectric. At a relatively low vibration frequency of electromagnetic field affecting the dielectric, this value is proportional to the electrical conductivity of the dielectric and inversely proportional to the frequency. In order to predict the expected value of the electric conductivity of the dielectric with metallic inclusions, a mathematical model that properly describes the structure of the composite and the electrical interaction of the matrix and inclusions is required.In the paper, a mathematical model of the electrical interaction of the representative element of the composite structure and a homogeneous isotropic medium with electrical conductivity, which is desired characteristics of the composite, is constructed. Globular shape of the metallic inclusions as an average statistical form of dispersed inclusions with a comparable size in all directions is adopted. The inclusion is covered with a globular layer of electrical insulation to avoid percolation with increasing volume concentration of inclusions. Outer globular layer of representative structure of composite elements consists of the dielectric material matrix.Quantitative analysis of two-sided estimates of possible values of the electrical conductivity of the composite, which are constructed by using dual variational electrokinetics problem statement for a heterogeneous solid body, showed that for real dielectric matrix material combinations and metallic inclusions in case when their electrical conductivity can differ by more than 10 orders of magnitude, these estimates can vary widely the specified characteristics of a composite. Therefore, to obtain the estimated effective dependence, a solution to the electrokinetics problem for representative element of the composite structure based on the assumption about ideal conductivity of metallic inclusions is found. It is shown that this dependence reflects properly the influence of the properties of the structural elements of a composite on its electrical conductivity.DOI: 10.7463/mathm.0315.079359