Account for the generalized derivative and the collective influence of phases on the homogenization process

A. V. Mishin

Sibirskij žurnal industrialʹnoj matematiki, Tome 25 (2022) no. 4, pp. 86-98.

Résumé

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 averaged Green's function based on the modified operator obtained and the averaging carried out. The solution has the form of the Yukawa potential, which characterizes from the physical point of view the transition layer caused by charge screening. It is a consequence of the integro-differential equation analysis with discontinuities and the introduced hypotheses. 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. The effective transport coefficients integrally take into account the microstructure of the system (physical properties of phases and characteristic scales) in an explicit form, which is a consequence of the found solution.

Détail

Keywords: heterogeneous medium, transition layer, generalized derivative, Green's function, averaging. .
Mots-clés : microstructure

Bibliographie

[1] L. P. Khoroshun, “A new mathematical model of the nonuniform deformation of composites”, Mekh. Kompos. Mater., 31:3 (1995), 310–318

[2] L. P. Khoroshun, “Mathematical models and methods of the mechanics of stochastic composites”, Appl. Mech., 30:10 (2000), 30–62

[3] Z. Hashin, S. Shtrikman, “On some variational principles in anisotropic and nonhomogeneous elasticity”, J. Mech. Phys. Solids, 10:4 (1962), 335–342 | DOI | MR

[4] Z. Hashin, S. Shtrikman, “A variational approach to the theory of the elastic behavior of multiphase materials”, J. Mech. Phys. Solids, 11:2 (1963), 127–140 | DOI | MR

[5] Z. Hashin, S. Shtrikman, “Conductivity of polycrystals”, Phys. Rev., 130:129 (1963), 129–133 | DOI

[6] Z. Khashin, S. Shtrikman, “Variatsionnyi podkhod k teorii uprugogo povedeniya mnogofaznykh materialov”, Zhurn. mekhaniki i fiziki tverdogo tela, 11:2 (1963), 127–140

[7] D. A.G. Bruggeman, “Berechnung verschiedener physikalischer Konstanten von heterogenen Substanzen. II. Dielektrizit-atskonstanten und Leitf-ahigkeiten von Vielrkistallen der nichtregularen Systeme”, Ann. Phys., 417:25 (1936), 645–672 | DOI

[8] E. Kroner, “Berechnung der elastischen Konstanten des Vielkristalls aus den Konstanten des Einkristalls”, Z. Phys., 151:4 (1958), 504–518 | DOI

[9] R. A. Hill, “Self-consistent mechanics of composite materials”, J. Mech. Phys. Solids, 13:4 (1965), 213–222 | DOI

[10] R. M. Christensen, Theory of Viscoelasticity, Acad. Press, N. Y., 1982

[11] J. Eshelby, Continuum Theory of Dislocations, Acad. Press, N.Y., 1956

[12] L. P. Khoroshun, “Random functions theory in problems on the macroscopic characteristics of microinhomogeneous media”, Appl. Mech., 14 (1978), 3–17

[13] L. P. Khoroshun, “Conditional-moment method in problems of the mechanics of composite materials”, Appl. Mech., 23:10 (1987), 100–108

[14] V. V. Bolotin, V. N. Moskalenko, “Opredelenie uprugikh postoyannykh mikroneodnorodnoi sredy”, Zhurn. prikl. mekhaniki i tekhn. fiziki, 34:1 (1968), 66–72

[15] R. I. Nigmatulin, Osnovy mekhaniki geterogennykh sred, Nauka, M., 1978 | MR

[16] N. S. Bakhvalov, G. P. Panasenko, Osrednenie protsessov v periodicheskikh sredakh, Nauka, M., 1984 | MR

[17] T. Mori, K. Tanaka, “Average stress in matrix and average elastic energy of materials with misfitting inclusions”, Acta Metall, 21 (1973), 571–574 | DOI

[18] P. P. Castaneda, J. R. Willis, “The effect of spatial distribution on the effective behavior of composite materials and cracked media”, J. Mech. Phys. Solids, 43 (1995), 1919–1951 | DOI | MR

[19] A. Fedotov, “The hybrid homogenization model of elastic anisotropic porous materials”, J. Mater. Sci., 53 (2018), 5092–5102 | DOI

[20] A. V. Mishin, “Obobschennaya proizvodnaya i ee ispolzovanie dlya analiza mikrostruktury geterogennoi sredy”, Sib. zhurn. industr. matematiki, 24:4 (2021), 79–96 | DOI

[21] J. Gao et al, “Networks formed from interdependent networks”, Nat. Phys., 8:1 (2012), 40–48 | DOI

[22] X. Huang et al, “Robustness of interdependent networks under targeted attack”, Phys. Rev. E. Stat. Nonlinear, Soft Matter Phys., 83:6 (2011), 065101 | DOI

[23] J. Gao et al, “Robustness of a network of networks”, Phys. Rev. Lett., 107:19 (2011), 195701 | DOI

[24] L. Shvarts, Matematicheskie metody dlya fizicheskikh nauk, Mir, M., 1965

[25] L. P. Khoroshun, “Elastic properties of materials reinforced with unidirectional short fibers”, Appl. Mech., 8:10 (1972), 1–7