Carrying out homogenization in viscoelastic heterogeneous media taking into account the collective influence of boundaries

A. V. Mishin

Sibirskij žurnal industrialʹnoj matematiki, Tome 26 (2023) no. 1, pp. 98-107.

Résumé

The effective coefficients of viscoelasticity of a heterogeneous medium are obtained on the basis of the generalized derivative formalism, which reflects the internal boundaries of a heterogeneous medium. A solution is sought for the averaged Green's function for the found modified operator, taking into account the averaging and subsequent analysis of the operator. The effective viscoelasticity coefficients integrally take into account the microstructure of the system (physical properties and characteristic phase sizes) in an explicit form, that is a consequence of the solution obtained, which expresses the solution of the many-body problem in a heterogeneous medium.

Détail

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

Bibliographie

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

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

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

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

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

[6] 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 | Zbl

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

[8] T. D. Shermergor, Teoriya uprugosti mikroneodnorodnykh sred, Nauka, M., 1977

[9] 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

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

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

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

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

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

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

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

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

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

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

[20] 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 | Zbl

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

[22] L. Y. Chen et al, “Processing and properties of magnesium containing a dense uniform dispersion of nanoparticles”, Nature, 528 (2015), 7583 | DOI

[23] K. Morsi, V. V. Patel, “Processing and properties of titanium-titanium boride (TiBw) matrix composites”, J. Mater. Sci., 42:6 (2007), 2037–2047 | DOI

[24] A. V. Mishin, V. M. Fomin, “Investigation of the elastic properties of the material obtained by the method of cold gas-dynamic spraying with laser treatment”, Appl. Mech. Tech. Phys., 2021, no. 6, 966–971 | DOI

[25] V. M. Fomin et al, “Deposition of cermet coatings on the basis of Ti, Ni, WC, and B4C by cold gas dynamic spraying with subsequent laser irradiation”, Phys. Mesomech., 23:4 (2020), 291–300 | DOI

[26] Q. An et al, “Two-scale TiB/Ti64 composite coating fabricated by two-step process”, J. Alloys Compd., 755 (2018), 29–40 | DOI

[27] V. M. Fomin, A. A. Golyshev, A. G. Malikov i dr, “Sozdanie funktsionalno-gradientnogo materiala metodom additivnogo lazernogo splavleniya”, Prikl. matematika i teor. fizika, 61:5 (2020), 224–234

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

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

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

[31] A. V. Mishin, “Uchet obobschennoi proizvodnoi i kollektivnogo vliyaniya faz na protsess gomogenizatsii”, Sib. zhurn. industr. matematiki, 25:4 (2022), 86–98

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