Refinement of the Maxwell formula for composite reinforced by circular cross-section fibers. I: Using the Schwarz alternating method. (English) Zbl 1457.74167
Summary: The effective properties of the fiber-reinforced composite materials with fibers of circular cross section are investigated. The novel estimation for the effective coefficient of thermal conductivity refining the classical Maxwell formula is derived. The method of asymptotic homogenization is used. For analytical solution of the periodically repeated cell problem, the Schwarz alternating process is employed. The principal term of the refined formula coincides with the classical Maxwell formula. On the other hand, the refined formula can be used far beyond the area of applicability of the Maxwell formula. It can be used for dilute and non-dilute composites. It is confirmed by comparison with known numerical and asymptotic results.
MSC:
| 74Q15 | Effective constitutive equations in solid mechanics |
| 74E30 | Composite and mixture properties |
| 74F05 | Thermal effects in solid mechanics |
References:
| [1] | Maxwell, JC, Treatise on Electricity and Magnetism (1873), Oxford: Clarendon Press, Oxford · JFM 05.0556.01 |
| [2] | Landauer, R., The electrical resistance of binary metallic mixture, J. Appl. Phys., 23, 7, 779-784 (1952) · doi:10.1063/1.1702301 |
| [3] | Lichtenecker, K., Die Dielektrizitätskonstante natürlicher und künstlicher Mischkörper, Phys. Zeits., 27, 4, 115-158 (1926) · JFM 52.0914.17 |
| [4] | Shvidler, MI, Statistical Hydrodynamics of Porous Media (1985), Moscow: Nedra, Moscow |
| [5] | Milton, GW, The Theory of Composites (2002), Cambridge: Cambridge University Press, Cambridge · Zbl 0993.74002 |
| [6] | Torquato, S., Random Heterogeneous Materials: Microstructure and Macroscopic Properties (2002), New York: Springer, New York · Zbl 0988.74001 |
| [7] | McPhedran, RC; Poladian, L.; Milton, GW, Asymptotic studies of closely spaced, highly conducting cylinders, Proc. R. Soc. A, 415, 185-196 (1988) |
| [8] | Keller, JB, A theorem on the conductivity of a composite medium, J. Math. Phys., 5, 4, 548-549 (1964) · Zbl 0129.44001 · doi:10.1063/1.1704146 |
| [9] | Gluzman, S.; Mityushev, V.; Nawalaniec, W., Computational Analysis of Structured Media (2017), Amsterdam: Elsevier, Amsterdam · Zbl 1302.80016 |
| [10] | Drygaś, P.; Gluzman, S.; Mityushev, V.; Nawalaniec, W., Applied Analysis of Composite Media. Analytical and Computational Results for Materials Scientists and Engineers (2020), Amsterdam: Elsevier, Amsterdam |
| [11] | Christensen, RM, Mechanics of Composite Materials (2005), Mineola: Dover Publications, Mineola |
| [12] | Andrianov, IV; Kalamkarov, AL; Starushenko, GA, Analytical expressions for effective thermal conductivity of composite materials with inclusions of square cross-section, Compos. Part B Eng., 50, 44-53 (2013) · doi:10.1016/j.compositesb.2013.01.023 |
| [13] | Andrianov, IV; Kalamkarov, AL; Starushenko, GA, Three-phase model for a fiber-reinforced composite material, Compos. Struct., 95, 95-104 (2013) · doi:10.1016/j.compstruct.2012.07.003 |
| [14] | Kalamkarov, AL; Andrianov, II; Starushenko, GA, Refinement of the Maxwell formula for fibre-reinforced composites, J. Multiscale Model., 10, 1, 1950001-1-1950001-33 (2019) |
| [15] | Berdichevsky, VL, Variational Principles of Mechanics of Continuous Media (1983), Moscow: Nauka, Moscow · Zbl 0526.73027 |
| [16] | Mityushev, V.; Rylko, N., Maxwells approach to effective conductivity and its limitations, Q. J. Mech. Appl. Math., 66, 2, 241-251 (2013) · Zbl 1291.74156 · doi:10.1093/qjmam/hbt003 |
| [17] | Berlyand, L.; Mityushev, V., Generalized Clausius-Mossotti formula for random composite with circular fibers, J. Stat. Phys., 102, 1-2, 115-145 (2001) · Zbl 1072.82586 · doi:10.1023/A:1026512725967 |
| [18] | Levin, V.; Kanaun, S.; Markov, M., Generalized Maxwell’s scheme for homogenization of poroelastic composites, Int. J. Eng. Sci., 61, 75-86 (2012) · Zbl 1423.74276 · doi:10.1016/j.ijengsci.2012.06.011 |
| [19] | Mityushev, V., Transport properties of double-periodic arrays of circular cylinders, ZAMM, 77, 2, 115-120 (1997) · Zbl 0900.76243 · doi:10.1002/zamm.19970770209 |
| [20] | Mityushev, V., Cluster method in composites and its convergence, Appl. Math. Lett., 77, 44-48 (2018) · Zbl 1462.74041 · doi:10.1016/j.aml.2017.10.001 |
| [21] | Mityushev, V.; Drygas, P., Effective properties of fibrous composites and cluster convergence, Multiscale Model. Simul., 17, 2, 696-715 (2019) · Zbl 1421.74086 · doi:10.1137/18M1184278 |
| [22] | Mityushev, V., Random 2D composites and the generalized method of Schwarz, Adv. Math. Phys., 2015, 15 (2015) · Zbl 1375.74007 · doi:10.1155/2015/535128 |
| [23] | Mityushev, V.; Nawalaniec, W., Effective conductivity of a random suspension of highly conducting spherical particles, Appl. Math. Model., 72, 230-246 (2019) · Zbl 1481.74180 · doi:10.1016/j.apm.2019.03.020 |
| [24] | Schwarz, HA, Über einige Abbildungsaufgaben, Ges. Math. Abh., II, 11, 65-83 (1869) |
| [25] | Courant, R., Geometrische Funktionentheorie (1963), Berlin: Springer, Berlin |
| [26] | Kantorovich, LV; Krylov, VI, Approximate Methods of Higher Analysis (1958), Groningen: Noordhoff, Groningen · Zbl 0083.35301 |
| [27] | Mikhlin, SG, Integral Equations (1964), Oxford: Pergamon Press, Oxford |
| [28] | Abramowitz, M.; Stegun, IA, Handbook of Mathematical Functions, with Formulas, Graphs, and Mathematical Tables (1965), New York: Dover Publications, New York |
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