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The effective thermal conductivity of composites with interfaces oscillating in two directions around a curved surface. (English) Zbl 1457.74166

Summary: In many situations of practical and/or theoretical interest, the assumption that the interfaces between constituent phases of a composite are smooth is no longer appropriate, and the consideration of rough interfaces at microscopic scale is necessary. However, in micromechanics, when the interfaces between the constituent phases of composites become rough, all classical well-known micromechanical schemes resorting to Eshelby’s formalism cannot be applicable and the problem of determining the effective properties of composites become largely open. The present work aims to determine the effective thermal conductivity of a composite in which the interfaces between its constituent phases are perfectly bonded but oscillate quickly around a curved surface and along two directions. To achieve this objective, a two-scale homogenization method is proposed. In the first-scale homogenization, or microscopic-to-mesoscopic upscaling, the interfacial zone in which the interface oscillates is homogenized as an equivalent interphase by applying an asymptotic analysis. The thermal properties of the equivalent interphase can generally be determined by using a numerical approach based on the fast Fourier transform method. In particular case where the equivalent interphase is very thin, this interphase is then replaced with a general imperfect interface situated at its middle surface. By applying the equivalent inclusion method, every inclusion with imperfect interface is further substituted by an equivalent inclusion with perfect interface. In the second-scale homogenization, or mesoscopic-to-macroscopic upscaling, due to the fact that the interfaces are perfect, the effective thermal conductivity can be analytically obtained by using some well-known classical micromechanical schemes. To illustrate the two-scale homogenization method proposed in this work, the case of a layered composite with rough interfaces oscillating in two directions around a plane surface and the example of a composite cylinder with rough interface oscillating in two directions around a circumferential surface are studied in detail. The analytical or semi-analytical results given by the proposed two-scale homogenization method are shown to be in good agreement with the numerical ones provided by the finite element method and to comply with the Reuss, Voigt and Hashin-Shtrikman bounds.

MSC:

74Q10 Homogenization and oscillations in dynamical problems of solid mechanics
74E30 Composite and mixture properties
74F05 Thermal effects in solid mechanics
74A50 Structured surfaces and interfaces, coexistent phases
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[1] Kohler, W.; Papanicolaou, G.; Varadhan, S.; Chow, PL; Kohler, WE; Papanicolaou, G., Boundary and interface problems in regions with very rough boundaries, Multiple Scattering and Waves in Random Media, 165-197 (1981), Amsterdam: North-Holland, Amsterdam · Zbl 0467.73142
[2] Nevard, J.; Keller, J., Homogenization of rough boundaries and interfaces, SIAM J. Appl. Math., 57, 6, 1660-1686 (1997) · Zbl 1049.35502
[3] Kristensson, G., Homogenization of corrugated interfaces in electromagnetics, Prog. Electromagn. Res., 55, 1-31 (2005)
[4] Vinh, PC; Tung, DX, Homogenized equations of the linear elasticity in two-dimensional domains with very rough interfaces, Mech. Res. Commun., 37, 3, 285-288 (2010) · Zbl 1272.74044
[5] Vinh, PC; Tung, DX, Homogenization of rough two-dimensional interfaces separating two anisotropic solids, ASME J. Appl. Mech., 78, 4, 041014 (2011)
[6] Vinh, PC; Tung, DX, Homogenized equations of the linear elasticity theory in two-dimensional domains with interfaces highly oscillating between two circles, Acta Mech., 218, 3, 333-348 (2011) · Zbl 1398.74263
[7] Le-Quang, H.; He, Q-C; Le, H-T, Multiscale homogenization of elastic layered composites with unidirectionally periodic rough interfaces, Multiscale Model. Simul., 11, 4, 1127-1148 (2013) · Zbl 1302.74137
[8] Le, H-T; Le-Quang, H.; He, Q-C, The effective elastic moduli of columnar composites made of cylindrically anisotropic phases with rough interfaces, Int. J. Solids Struct., 51, 14, 2633-2647 (2014)
[9] Nguyen, D-H; Le, H-T; Le-Quang, H.; He, Q-C, Determination of the effective conductive properties of composites with curved oscillating interfaces by a two-scale homogenization procedure, Comput. Mater. Sci., 94, 150-162 (2014)
[10] Le-Quang, H.; Le, H-T; Nguyen, D-H; He, Q-C, Two-scale homogenization of elastic layered composites with interfaces oscillating in two directions, Mech. Mater., 75, 60-72 (2014)
[11] Nguyen, D.-H., Le-Quang, H., He, Q.-C., Tran, A.-T.: Generalized Hill-Mendel’s lemma and equivalent inclusion method for determining the effective thermal conductivity of composites with imperfect interfaces (submitted)
[12] Milton, G., The Theory of Composites (2002), Cambridge: Cambridge University Press, Cambridge · Zbl 0993.74002
[13] He, Q-C; Feng, Z-Q, Homogenization of layered elastoplastic composites: theoretical results, Int. J. Nonlinear Mech., 47, 2, 367-376 (2012)
[14] Papanicolaou, G.; Bensoussan, A.; Lions, J-L, Asymptotic Analysis for Periodic Structures (1978), Amsterdam: North-Holland, Amsterdam · Zbl 0411.60078
[15] Sanchez-Palencia, E., Non-homogeneous Media and Vibration Theory. Lecture Note in Physics (1980), Berlin: Springer, Berlin · Zbl 0432.70002
[16] Bakhvalov, N.; Panasenko, G., Averaging of Processes in Periodic Media: Mathematical Problems of the Mechanics of Composite Materials (1989), Dordrecht: Kluwer Acadamic Publishers, Dordrecht · Zbl 0692.73012
[17] Persson, L.; Svansted, N.; Wyller, J., The Homogenization Method: An Introduction (1993), Lund, Sweden: Studentlitterature, Lund, Sweden · Zbl 0847.73003
[18] Jikov, VV; Koslov, SM; Oleinik, OA, Homogenization of Differential Operators and Integral Functionals (1994), Berlin: Springer, Berlin · Zbl 0838.35001
[19] Moulinec, H.; Suquet, P., A fast numerical method for computing the linear and nonlinear mechanical properties of composites, C. R. Acad. Sci. Paris, 318, 1417-1423 (1994) · Zbl 0799.73077
[20] Moulinec, H.; Suquet, P., A numerical method for computing the overall response of nonlinear composites with complex microstructure, Comput. Methods Appl. Mech. Eng., 157, 1, 69-94 (1998) · Zbl 0954.74079
[21] Bonnet, G., Effective properties of elastic periodic composite media with fibers, J. Mech. Phys. Solids, 55, 5, 881-899 (2007) · Zbl 1170.74042
[22] Le-Quang, H.; Phan, T-L; Bonnet, G., Effective thermal conductivity of periodic composites with highly conducting imperfect interfaces, Int. J. Therm. Sci., 50, 8, 1428-1444 (2011)
[23] Sanchez-Palencia, E., Comportement limite d’un problème de transmission à travers une plaque mince et faiblement conductrice, C. R. l’Acad. Sci. Sér. A, 270, 1026-1028 (1970) · Zbl 0189.41503
[24] Pham-Huy, H.; Sanchez-Palencia, E., Phénomènes de transmission à travers des couches minces de conductivité élevée, J. Math. Anal. Appl., 47, 284-309 (1974) · Zbl 0286.35007
[25] Miloh, T.; Benveniste, Y., On the effective conductivity of composites with ellipsoidal inhomogeneities and highly conducting interfaces, Proc. R. Soc. A Math. Phys. Eng. Sci., 455, 2687-2706 (1999) · Zbl 0937.74051
[26] Hashin, Z., Thin interphase/imperfect interface in conduction, J. Appl. Phys., 89, 4, 2261-2267 (2001)
[27] Gu, ST; Monteiro, E.; He, Q-C, Coordinate-free derivation and weak formulation of a general imperfect interface model for thermal conduction in composites, Compos. Sci. Technol., 71, 9, 1209-1216 (2011)
[28] Le-Quang, H.; Pham, DC; Bonnet, G.; He, Q-C, Estimations of the effective conductivity of anisotropic multiphase composites with imperfect interfaces, Int. J. Heat Mass Transf., 58, 1, 175-187 (2013)
[29] Le-Quang, H., Determination of the effective conductivity of composites with spherical and spheroidal anisotropic particles and imperfect interfaces, Int. J. Heat Mass Transf., 95, 162-183 (2016)
[30] Bonfoh, N.; Dreistadt, C.; Sabar, H., Micromechanical modeling of the anisotropic thermal conductivity of ellipsoidal inclusion-reinforced composite materials with weakly conducting interfaces, Int. J. Heat Mass Transf., 108, 1727-1739 (2017)
[31] Bonfoh, N.; Sabar, H., Anisotropic thermal conductivity of composites with ellipsoidal inclusions and highly conducting interfaces, Int. J. Heat Mass Transf., 118, 498-509 (2018)
[32] Le-Quang, H., Estimations and bounds of the effective conductivity of composites with anisotropic inclusions and general imperfect interfaces, Int. J. Heat Mass Transf., 99, 327-343 (2016)
[33] Gu, S-T; Wang, A-L; Xu, Y.; He, Q-C, Closed-form estimates for the effective conductivity of isotropic composites with spherical particles and general imperfect interfaces, Int. J. Heat Mass Transf., 83, 317-326 (2015)
[34] Le-Quang, H.; Bonnet, G.; He, Q-C, Size-dependent eshelby tensor fields and effective conductivity of composites made of anisotropic phases with highly conducting imperfect interfaces, Phys. Rev. B, 81, 064203 (2010)
[35] Hashin, Z., Assessment of the self consistent scheme approximation: Conductivity of particulate composites, J. Compos. Mater., 2, 3, 284-300 (1968)
[36] Hashin, Z., Analysis of composite materials—a survey, J. Appl. Mech., 50, 3, 481-505 (1983) · Zbl 0542.73092
[37] Norris, AN, A differential scheme for the effective moduli of composites, Mech. Mater., 4, 1, 1-16 (1985)
[38] Kerner, EH, The elastic and thermo-elastic properties of composite media, Proc. Phys. Soc. Lond. Sect. B, 69, 8, 808-813 (1956)
[39] der Poel, CV, On the rheology of concentrated suspension, Rheol. Acta, 1, 198-205 (1958)
[40] Smith, JC, Correction and extension of van der poel’s method for calculating the shear modulus of a particulate composite, J. Res. Natl. Bureau Stand., 78A, 355-361 (1974)
[41] Christensen, RM; Lo, KH, Solutions for effective shear properties in three phase sphere and cylinder models, J. Mech. Phys. Solids, 27, 4, 315-330 (1979) · Zbl 0419.73007
[42] Chen, T.; Kuo, H-Y, Transport properties of composites consisting of periodic arrays of exponentially graded cylinders with cylindrically orthotropic materials, J. Appl. Phys., 98, 3, 033716 (2005)
[43] Rayleigh, L., On the influence of obstacles arranged in rectangular order upon the properties of a medium, Phil. Mag., 34, 481-502 (1892) · JFM 24.1015.02
[44] Nicorovici, NA; McPhedran, RC; Milton, GW, Transport properties of a three-phase composite material: the square array of coated cylinders, Proc. R. Soc. Lond. A, 442, 1916, 599-620 (1993)
[45] Hashin, Z.; Rosen, BW, The elastic moduli of fiber-reinforced materials, J. Appl. Mech., 31, 1964, 223-323 (1964)
[46] Hashin, Z., The elastic moduli of heterogeneous materials, J. Appl. Mech., 29, 1, 143-150 (1962) · Zbl 0102.17401
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