Abstract
The effective transfer coefficients of a heterogeneous medium are obtained based on the formalism of a generalized derivative, which reflects the internal boundaries of a heterogeneous medium. The formula for the generalized derivative is a consequence of applying the variational apparatus to the energy functional for a heterogeneous medium, taking into account the indicator function characterizing the phase at a point. The solution is sought for the homogenized Green’s function based on the modified operator obtained, and the homogenization is carried out. Based on an analysis of an integro-differential equation with discontinuities and the introduced hypotheses, the solution has the form of a Yukawa potential that characterizes the transition layer caused by charge screening from the physical point of view. This potential is aimed at expressing the solution of the many-body problem in a heterogeneous medium and reflecting the collective influence of the phases on the field propagating through the system. As a result of the solution found, the effective transport coefficients integrally take into account the microstructure of the system (physical properties of phases and characteristic scales) in an explicit form.
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The work was supported financially by the Ministry of Education and Science of the Russian Federation, project no. 075-15-2020-781.
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Translated by V. Potapchouck
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Mishin, A.V. Taking into Account the Generalized Derivative and the Collective Influence of Phases on the Homogenization Process. J. Appl. Ind. Math. 16, 684–694 (2022). https://doi.org/10.1134/S1990478922040093
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DOI: https://doi.org/10.1134/S1990478922040093