A Variational Approach to the Estimate of the Permittivity of a Composite with Dispersed Inclusions
<p>Composites are inhomogeneous materials (heterogeneous solid body, which fall into the matrix and inclusions. The matrix in a composite is a binder between the inclusions. The properties of the inclusions mainly determine the application of composites. Selection of the characteristics of the...
| Main Authors: | , , |
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| Format: | Article |
| Language: | Russian |
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MGTU im. N.È. Baumana
2015-01-01
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| Series: | Matematika i Matematičeskoe Modelirovanie |
| Subjects: | |
| Online Access: | http://mathm.elpub.ru/jour/article/view/15 |
| _version_ | 1811333517330612224 |
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| author | V. S. Zarubin G. N. Kuvyrkin O. V. Pugachev |
| author_facet | V. S. Zarubin G. N. Kuvyrkin O. V. Pugachev |
| author_sort | V. S. Zarubin |
| collection | DOAJ |
| description | <p>Composites are inhomogeneous materials (heterogeneous solid body, which fall into the matrix and inclusions. The matrix in a composite is a binder between the inclusions. The properties of the inclusions mainly determine the application of composites. Selection of the characteristics of the matrix and inclusions enables us to meet the requirements for materials to be used in various fields of technology. Composites are widely used as structural or thermal protection material and as functional materials in various electrical devices, including dielectrics. One of the most important characteristics of the composite dielectric is the relative permittivity. The latter is primarily determined by the dielectric properties of the matrix and inclusions, as well as the shape and volume concentration of inclusions.<br />For a composite with dispersed inclusions we are able to construct adequate mathematical models which enable us to predict sufficiently reliably the dependence of its dielectric constant on these defining parameters. In this paper, among the various approaches to the construction of such models we emphasize a variational approach which allows us not only to determine this dependence, but also obtain guaranteed bilateral boundaries of the area of possible values of the dielectric constant of the composite used to estimate the highest accuracy of calculated values.<br />The representative element of the composite structure with inclusions of spherical shape modeling the form of dispersed inclusions with dimensions close to all directions is considered. For the representative element we obtained the electrostatic potential distribution that is permissible for the minimized functional. The latter is the part of the variational form of a mathematical model which describes the dielectric properties of the considered composite. From the equality of the values of this functional on the received permissible distribution in a representative element of the composite structure and on the distribution in the equal-element of homogeneous medium with the desired dielectric constant of the composite we determined the dependence of this value on the dielectric characteristics of the matrix and inclusions and on the volume concentration of inclusions.<br />Quantitative analysis of the obtained dependence in a wide range of defining parameters showed that all the results of the calculations are located in the area of possible values. This area is defined by constructed bilateral estimates. This confirms the appropriate use of the variational approach and the possibility of its application for the prediction of the dielectric characteristics of composites with dispersed inclusions.</p> |
| first_indexed | 2024-04-13T16:52:49Z |
| format | Article |
| id | doaj.art-481b08d597a943f09067ddb7b62f7556 |
| institution | Directory Open Access Journal |
| issn | 2412-5911 |
| language | Russian |
| last_indexed | 2024-04-13T16:52:49Z |
| publishDate | 2015-01-01 |
| publisher | MGTU im. N.È. Baumana |
| record_format | Article |
| series | Matematika i Matematičeskoe Modelirovanie |
| spelling | doaj.art-481b08d597a943f09067ddb7b62f75562022-12-22T02:38:53ZrusMGTU im. N.È. BaumanaMatematika i Matematičeskoe Modelirovanie2412-59112015-01-0102374914A Variational Approach to the Estimate of the Permittivity of a Composite with Dispersed InclusionsV. S. Zarubin0G. N. Kuvyrkin1O. V. Pugachev2Bauman Moscow State Technical University, RussiaBauman Moscow State Technical University, RussiaBauman Moscow State Technical University, Russia<p>Composites are inhomogeneous materials (heterogeneous solid body, which fall into the matrix and inclusions. The matrix in a composite is a binder between the inclusions. The properties of the inclusions mainly determine the application of composites. Selection of the characteristics of the matrix and inclusions enables us to meet the requirements for materials to be used in various fields of technology. Composites are widely used as structural or thermal protection material and as functional materials in various electrical devices, including dielectrics. One of the most important characteristics of the composite dielectric is the relative permittivity. The latter is primarily determined by the dielectric properties of the matrix and inclusions, as well as the shape and volume concentration of inclusions.<br />For a composite with dispersed inclusions we are able to construct adequate mathematical models which enable us to predict sufficiently reliably the dependence of its dielectric constant on these defining parameters. In this paper, among the various approaches to the construction of such models we emphasize a variational approach which allows us not only to determine this dependence, but also obtain guaranteed bilateral boundaries of the area of possible values of the dielectric constant of the composite used to estimate the highest accuracy of calculated values.<br />The representative element of the composite structure with inclusions of spherical shape modeling the form of dispersed inclusions with dimensions close to all directions is considered. For the representative element we obtained the electrostatic potential distribution that is permissible for the minimized functional. The latter is the part of the variational form of a mathematical model which describes the dielectric properties of the considered composite. From the equality of the values of this functional on the received permissible distribution in a representative element of the composite structure and on the distribution in the equal-element of homogeneous medium with the desired dielectric constant of the composite we determined the dependence of this value on the dielectric characteristics of the matrix and inclusions and on the volume concentration of inclusions.<br />Quantitative analysis of the obtained dependence in a wide range of defining parameters showed that all the results of the calculations are located in the area of possible values. This area is defined by constructed bilateral estimates. This confirms the appropriate use of the variational approach and the possibility of its application for the prediction of the dielectric characteristics of composites with dispersed inclusions.</p>http://mathm.elpub.ru/jour/article/view/15compositepermittivitydispersed inclusions |
| spellingShingle | V. S. Zarubin G. N. Kuvyrkin O. V. Pugachev A Variational Approach to the Estimate of the Permittivity of a Composite with Dispersed Inclusions Matematika i Matematičeskoe Modelirovanie composite permittivity dispersed inclusions |
| title | A Variational Approach to the Estimate of the Permittivity of a Composite with Dispersed Inclusions |
| title_full | A Variational Approach to the Estimate of the Permittivity of a Composite with Dispersed Inclusions |
| title_fullStr | A Variational Approach to the Estimate of the Permittivity of a Composite with Dispersed Inclusions |
| title_full_unstemmed | A Variational Approach to the Estimate of the Permittivity of a Composite with Dispersed Inclusions |
| title_short | A Variational Approach to the Estimate of the Permittivity of a Composite with Dispersed Inclusions |
| title_sort | variational approach to the estimate of the permittivity of a composite with dispersed inclusions |
| topic | composite permittivity dispersed inclusions |
| url | http://mathm.elpub.ru/jour/article/view/15 |
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