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Subsequent Generalizations of Theory and Related Problems

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Local and Nonlocal Micromechanics of Heterogeneous Materials

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Abstract

In this chapter we obtain a unified operator form of the GIE for the general cases of local and nonlocal problems, static, and wave motion phenomena for composite materials with periodic and random structures containing inclusions with perfect and imperfect interfaces and subjected to any number of coupled or uncoupled, homogeneous, or inhomogeneous external fields of different physical nature. Estimations of the effective properties and both the first and second statistical moments of fields in the constituents of CMs are presented in a general form of perturbations introduced by the heterogeneities and taking into account a possible imperfection of interface conditions. Some particular cases, asymptotic representations, and simplifications are presented for linear thermoelastic cases, conductivity problem, problems for piezoelectric and other coupled phenomena, composites with nonlocal elastic properties of constituents, and the wave propagation in composites with electromagnetic, optic, and mechanical responses.

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Notes

  1. 1.

    For example, it is known that for 2D elastic problems the plane–strain state is only possible for material symmetry no lower than orthotropic (see e.g. [910] that will be assumed hereafter in 2D case.

References

  1. Acrivos A, Chang E (1986) A model for estimating transport quantities in two-phase materials. Phys Fluids, 29:3–4

    Article  Google Scholar 

  2. Avellaneda M (1987) Iterated homogenization, differential effective medium theory and applications. Commun Pure Appl Math, 40: 527–554

    Article  MathSciNet  MATH  Google Scholar 

  3. Axel F (1992) Bounds for field fluctuations in two-phase materials. J Appl Phys, 72:1217–1220

    Article  Google Scholar 

  4. Ballas J, Sladek J, Sladek V (1989) Stress Analysis by Boundary Element Methods. Elsevier, Amsterdam

    MATH  Google Scholar 

  5. Barnett DM, Lothe J (1975) Dislocation and line charges in anisotropic piezoelectric insulators. Phys Stat Solids (b), 67:105–111

    Article  Google Scholar 

  6. Batchelor GK (1974) Transport properties of two-phase materials with random structure. Ann-Rev Fluid Mech, 6:227–255

    Article  MATH  Google Scholar 

  7. Bazǎnt ZP, Jirásek M (2002) Nonlocal integral formulations of plasticity and damage Survey of progress. J. Engineering Mechanics, 128:1119–1149.

    Google Scholar 

  8. Benveniste Y (1986) On the effective thermal conductivity of multiphase composites. J Appl Math Phys (ZAMP), 37:696–713

    Article  MATH  Google Scholar 

  9. Benveniste Y (1987a) A new approach to application of Mori-Tanaka’s theory in composite materials. Mech Mater, 6:147–157

    Article  Google Scholar 

  10. Benveniste Y (2006) A general interface model for a three-dimensional curved thin anisotropic interphase between two anisotropic media. J Mech Phys Solids, 54:708–734

    Article  MathSciNet  MATH  Google Scholar 

  11. Benveniste Y, Dvorak GJ (1992) On uniform fields and universal relations in piezoelectric composites J Mech Phys Solids, 40:1295–1312

    Google Scholar 

  12. Benveniste Y, Miloh T (1986) The effective conductivity of composites with imperfect contact at constituent interfaces. Int J Eng Sci, 24:1537–1552

    Article  MATH  Google Scholar 

  13. Benveniste Y, Milton GW(2003) New exact results for the e’ective electric, elastic, piezoelectric and other properties of composite ellipsoid assemblages. J Mech Phys Solids, 51:1773 –1813

    Article  MathSciNet  MATH  Google Scholar 

  14. Beran M (1974) Application of statistical theories for the determination of thermal, electrical and magnetic properties of heterogeneous materials. In: Sendeckyj GP (ed), Mechanics of Composite Materials. Academic Press, New York, 2:209–249

    Google Scholar 

  15. Beran M (1980) Field fluctuations in a two-phase random medium. J Math Phys, 21:2583–2585

    Article  Google Scholar 

  16. Beran MJ, McCoy JJ (1970a) Mean field variations in a statistical sample of heterogeneous linearly elastic solids. Int J Solid Struct, 6:1035–1054

    Article  MATH  Google Scholar 

  17. Beran MJ, McCoy JJ (1970b) Mean field variation in random media. Quart Appl Math, 28:245–257

    Article  MathSciNet  MATH  Google Scholar 

  18. Bergman DJ (1978) The dielectric constant of a composite material — a problem of classical physics. Phys Rep, 43C:377–407

    Article  MathSciNet  Google Scholar 

  19. Brosseau C (2006) Modelling and simulation of dielectric heterostructures: a physical survey from an historical perspective. J Phys D: Appl Phys, 39:1277–1294

    Article  Google Scholar 

  20. Brown WF (1955) Solid mixture permittivities. J Phys Chem, 23:1514–1517

    Article  Google Scholar 

  21. Bruggeman DAG (1935) Berechnung verschiedener physikalischer Konstante von hetero-genete substanze I: Dielektrizitätskonstanten und leitfähigkeiten der misckörper aus isotropen substanzen. Annal Physik, 24:636–679

    Article  Google Scholar 

  22. Buevich YA (1992) Heat and mass transfer in disperse media. I. Average field equations. Int J Heat Mass Transfer, 35:2445–2452

    Article  MATH  Google Scholar 

  23. Buevich YA, Ustinov VA (1995) Effective conductivity of a macroscopically inhomogeneous dispersions. Int J Heat Mass Transfer, 38:381–389

    Article  MATH  Google Scholar 

  24. Buryachenko VA (1993c) Effective strength properties of elastic physically nonlinear composites. In: Marigo JJ, Rousselier G (eds), Proc. of the MECAMAT Conf. Micromechanics of Materials. Editions Eyrolles, Paris, 567–578

    Google Scholar 

  25. Buryachenko VA (2007) Generalization of the multiparticle effective field method in static of random structure matrix composites. Acta Mechan, 188:167–208

    Article  MATH  Google Scholar 

  26. Buryachenko VA (2007a) Generalization of the multiparticle effective field method in static of random structure matrix composites. Acta Mechanica, 188:167–208

    Article  MATH  Google Scholar 

  27. Buryachenko VA (2007b) Micromechanics of Heterogeneous Materials. Springer, NY

    Book  MATH  Google Scholar 

  28. Buryachenko VA (2010b) On the thermo-elastostatics of heterogeneous materials. I. General integral equation. Acta Mech, 213:359–374.

    Article  MATH  Google Scholar 

  29. Buryachenko VA (2010c) On the thermo-elastostatics of heterogeneous materials. II. Analyze and generalization of some basic hypotheses and propositions. Acta Mech, 213:375-398.

    Article  MATH  Google Scholar 

  30. Buryachenko VA (2011a) Inhomogeneity of the first and second statistical moments of stresses inside the heterogeneities of random structure matrix composites. Int. J. Solids and Structures, 48:1665–1687.

    Article  MATH  Google Scholar 

  31. Buryachenko VA (2011c) On thermoelastostatics of composites with nonlocal properties of constituents. II. Estimation of effective material and field parameters. Int. J. Solids and Structures, 48:1829–1845.

    Article  Google Scholar 

  32. Buryachenko VA (2013) General integral equations of micromechanics of composite materials with imperfectly bonded interfaces. Int. J. Solids and Structures, 50:3190–3206.

    Article  Google Scholar 

  33. Buryachenko V (2014c) Solution of general integral equations of micromechanics of heterogeneous materials. J. Solids and Structures, 51:3823–3843 (130 refs)

    Google Scholar 

  34. Buryachenko VA, Brun M (2011) FEA in elasticity of random structure composites reinforced by heterogeneities of noncanonical shape. Int. J. Solid Struct. 48:719–728.

    Article  MATH  Google Scholar 

  35. Buryachenko VA, Brun M (2012a) Random residual stresses in elasticity homogeneous medium with inclusions of noncanonical shape. Int. J. Multiscale Comput. Enging. 10:261-279.

    Article  Google Scholar 

  36. Buryachenko VA, Murov V A (1991) Effective conductivity of matrix composites. Inzhenerno Fiz Zhurnal, 61(2):305–312 (In Russian. Engl Transl. J Engng Phys, 61:1041–1047)

    Google Scholar 

  37. Buryachenko VA, Parton VZ (1990a) Effective Helmholtz operator for matrix composites. Izv AN SSSR, Mekh Tverd Tela (3):55–63 (In Russian. Engl Transl. Mech Solids, 25:60–69

    Google Scholar 

  38. Buryachenko VA, Parton VZ (1991) Effective parameters of static conjugating physical-mechanical fields in matrix composites. Fiziko-Khimichescaja Mech Mater, 27(4):105–111 (In Russian. Engl Transl. Sov Mater Sci, 27:428–433)

    Google Scholar 

  39. Buryachenko VA, Parton VZ (1992c) Effective strength parameters of composites in coupled physicomechanical fields. Priklad Mekh Tekhn Fiz, (4), 124–130 (In Russian. Engl Transl. J Appl Mech Tech Phys, 33:589–593)

    Google Scholar 

  40. Buryachenko VA, Rammerstorfer FG (1998a) Micromechanics and nonlocal effects in graded random structure matrix composites. In: Bahei-El-Din YA, Dvorak GJ (eds) IUTAM Symp. on Transformation Problems in Composite and Active Materials. Kluwer, Dordrecht, 197–206

    Google Scholar 

  41. Buryachenko VA, Rammerstorfer FG (2001) Local effective thermoelastic properties of graded random structure composites. Arch Appl Mech, 71:249–272

    Article  MATH  Google Scholar 

  42. Chekin BC (1970) Effective parameters of elastic medium with randomly distributed cracks. Izv AN SSSR, Fiz Zemli, N10:13–21 (In Russian. Engl Transl. Phys Solid Earth, 5)

    Google Scholar 

  43. Chen CH, Wang YC (1996) Effective thermal conductivity of misoriented short-fiber reinforced composites. Mech Mater, 23:217–228

    Article  Google Scholar 

  44. Chen P, Shen Y (2007) Propagation of axial shear magneto—electro–elastic waves in piezoelectric—piezomagnetic composites with randomly distributed cylindrical inhomogeneities. Int. J. Solids and Structures, 44:1511–1532.

    Article  MATH  Google Scholar 

  45. Chen T (1993b) Piezoelectric properties of multiphase fibrous composites: some theoretical results. J Mech Phys Solids, 41:1781–1794

    Article  MathSciNet  MATH  Google Scholar 

  46. Chen TY (1994) Micromechanical estimates of the overall thermoelectroelastic moduli of multiphase fibrous composites. Int J Solids Struct, 31:3099–3111

    Article  MATH  Google Scholar 

  47. Chen T (1999) Exact moduli and bounds of two-phase composites with coupled multifield linear responses J Mech Phys Solids, 45:385–398

    Google Scholar 

  48. Chen T, Dvorak GJ, Benveniste Y (1990) Stress fields in composites reinforced by coated cylindrically orthotropic fibers Mechanics of Materials, 9:17–32

    Google Scholar 

  49. Chen T, Dvorak GJ, Yu CC (2007) Size-dependent elastic properties of unidirectional nano-composites with interface stresses. Acta Mechan, 188:39–54

    Article  MATH  Google Scholar 

  50. Chen T, Dvorak GJ, Yu CC (2007) Solids containing spherical nano-inclusions with interface stresses effective properties and thermal-mechanical connections. Int. J. Solids and Structures, 44:941–955.

    Article  MATH  Google Scholar 

  51. Chen T, Yang S-H (1995) The problem of thermal conductivity for two ellipsoidal inhomogeneities in an anisotropic medium and its relevance to composite materials. Acta Mechan, 111:41–58

    Article  MATH  Google Scholar 

  52. Cheng H, Torquato S (1997) Electric-field fluctuations in random dielectric composites. Phys Rev, B56:8060–8068

    Article  Google Scholar 

  53. Chiew Y-C, Glandt ED (1987) Effective conductivity of dispersion: the effect of resistance at the particle surfaces. Chem Engng Sci, 42:2677–2685

    Article  Google Scholar 

  54. Choy TA, Alexandropoulos A, Thorpe MF (1998) Dielectric function for a material containing hyperspherical inclusions in O(c 2): I. Multipole expansions; II. Method of images. Proc Roy Soc Lond Ser, A454(1975):1973–1992, 1993–2013

    Google Scholar 

  55. Corcolle R, Préault V, Dani L (2012) Romain Corcolle, Valentin Préault, and Laurent Dani Second order moments in linear smart material composites. IEEE Transactions on Magnetics, 48:663-666

    Article  Google Scholar 

  56. Creswik RJ, Farah HA, Poole CP (1998) Introduction to Renormalization Group Methods Physics. John Wiley & Sons, New York

    Google Scholar 

  57. Datta SK (1977) A self-consistent approach to multiple scattering by elastic ellipsoidal inclusions. J Appl Mech, 44:657–661

    Article  Google Scholar 

  58. Deeg WF (1980) The Analysis of Dislocation, Crack and Inclusion Problems in Piezoelectric Solids. PhD Thesis, Stanford University, Stanford, CA

    Google Scholar 

  59. Ding C-L, Zhao X-P (2011) Multi-band and broadband acoustic metamaterial with resonant structures J. Phys. D Appl. Phys., 44:215402 (8pp.)

    Google Scholar 

  60. Dinzart F, Sabar H (2011) Magneto-electro-elastic coated inclusion problem and its application to magnetic-piezoelectric composite materials. Int. J. Solids and Structures, 48:2393–2401.

    Article  Google Scholar 

  61. Dinzart F, Sabar H (2017) New micromechanical modeling of the elastic behavior of composite materials with ellipsoidal reinforcements and imperfect interfaces. Int.l J. of Solids and Structures, 108:254-262

    Article  Google Scholar 

  62. Dinzart F, Sabar H. (2019) Electroelastic ellipsoidal inclusion with imperfect interface and its application to piezoelectric composite materials Int. J. Solids Structures, 136–137, 241–249

    Google Scholar 

  63. Djiordjevic BR, Hetherington JH, Thorpe, MF (1996) Spectral function for a conducting sheet containing circular inclusions. Phys Rev, B35:14862–14871

    Article  Google Scholar 

  64. Duan HL, Karihaloo BL (2007) Thermo-elastic properties of heterogeneous materials with imperfect interfaces Generalized Levin’s formula and Hill’s connections. J. Mechanics Physics Solids, 55:1037–1052.

    Article  MathSciNet  MATH  Google Scholar 

  65. Dul’nev GH, Malarev BI (1990) Percolation theory in the conductivity theory of inhomogeneous media. Inzhenerno Fiz Zhurnal, 39:522–539 (In Russian)

    Google Scholar 

  66. Dumont , Serpilli M, Rizzoni R, Lebon FC (2020) Numerical validation of multiphysic imperfect interfaces models. Frontiers in Materials, 2020, 7:158 (14 pages)

    Google Scholar 

  67. Dunn ML(1993) Exact relations between the thermoelectroelastic moduli of heterogeneous materials. Proc Roy Soc Lond, A441:549–557

    MathSciNet  MATH  Google Scholar 

  68. Dunn ML (1994a) Electroelastic Green’s functions for transversely isotropic piezoelectric media and their applications to the solutions of inclusion and inhomogeneity problems. Int J Engrg Sci, 32:119–131

    Article  MathSciNet  MATH  Google Scholar 

  69. Dunn ML (1994b) Thermally induced Relds in electroelastic composite materials: average Relds and effective behavior. J Engng Mat Technol, 116:200–207

    Article  Google Scholar 

  70. Dunn ML, Taya M (1993a) Micromechanics predictions of the effective electroelastic moduli of piezoelectric composites. Int J Solids Struct, 30:161-175

    Article  MATH  Google Scholar 

  71. Dunn ML, Taya M (1993b) An analysis of piezoelectric composite materials containing ellipsoidal inhomogeneities. Proc R Soc Lond: A 443: 265–287

    Google Scholar 

  72. Dvorak GJ, Benveniste Y (1992a) On the thermomechanics of composites with imperfectly bonded interfaces and damage. Int J Solids Struct, 29:2907–2919

    Article  MATH  Google Scholar 

  73. Dykhne AM (1970) Conductivity of a two-dimensional two-phase system. J Experiment Theor Phys (JETP) 59:110–116 (In Russian. Engl Trunsl. Soviet Phys, (1971) 32:63–65)

    Google Scholar 

  74. Edelen DGB, Laws N (1971) On the thermodynamics of systems with nonlocality. Arch. Ration. Mech. Anal., 43:36–44

    Article  MathSciNet  MATH  Google Scholar 

  75. Eringen AC (1999) Microcontinuum Field Theories I. Foundations and Solids. Springer-Verlag, Berlin

    Book  MATH  Google Scholar 

  76. Eringen AC (2002) Nonlocal Continuum Field Theories. Springer-Verlag, New York

    MATH  Google Scholar 

  77. Eshelby JD (1957) The determination of the elastic field of an ellipsoidal inclusion, and related problems. Proc Roy Soc Lond, A241:376–396

    MathSciNet  MATH  Google Scholar 

  78. Eshelby JD (1961) Elastic inclusion and inhomogeneities. In: Sneddon IN, Hill R (eds), Prog in Solid Mechan. North-Holland, Amsterdam, 2:89–140

    Google Scholar 

  79. Filatov AN, Sharov LV (1979) Integral Inequalities and the Theory of Nonlinear Oscillations. Nauka, Moscow (In Russian)

    Google Scholar 

  80. Fish J, Filonova V, Kuznetsov S (2012b) Micro-inertia effects in nonlinear heterogeneous media Int. J. Numer. Meth. Engng, 91:1406–1426

    Google Scholar 

  81. Fokin AG, Shermergor TD (1989) Theory of propagation of elastic waves in nonhomogeneous media. Mekhan. Kompoz. Materialov, 25(5), 821–832 (In Russian. Engl Transl. Mech. Compos. Materials, (1990) 25:600–609.)

    Google Scholar 

  82. Fu LS (1987) Dynamic moduli and located damage in composites. In: Vary A (ed), Material Analysis by Ultrasonics: Metals, Ceramics, Composites. Noyes Data Corp, New York, 225–248.

    Google Scholar 

  83. Fu LS, Mura T (1983) The determination of elastodynamic fields of an ellipsoidal inhomogeneity. J Appl Mech, 50:390–396

    Article  MATH  Google Scholar 

  84. Furmañski P (1997) Head conduction in composites: homogenization and macroscopic behavior. Appl Mech Rev, 50:327–356

    Article  Google Scholar 

  85. Gao X, Huang Z, Fang D (2017) Curvature-dependent interfacial energy and its effects on the elastic properties of nanomaterials. Int. J. Solids Structures, 113–114:100–107

    Article  Google Scholar 

  86. Gel’fand IA, Shilov G (1964) Generalized Functions. Academic Press, 1, New York

    Google Scholar 

  87. Ghosh S (2011) Micromechanical Analysis and Multi-Scale Modeling Using the Voronoi Cell Finite Element Method (Computational Mechanics and Applied Analysis). CRC Press, Boca Raton

    MATH  Google Scholar 

  88. Gibiansky L, Torquato S (1998) New approximation for the effective energy of nonlinear conducting composites. J Appl Phys, 84:301–305

    Article  Google Scholar 

  89. Giordano S (2005) Order and disorder in heterogeneous material microstructure: electric and elastic characterisation of dispersion of pseudo-oriented spheroids. Int J Engng Sci, 43:1033–1058

    Article  MATH  Google Scholar 

  90. Gordon JA, Ziolkowski RW (2008) CNP optical metamaterials. Optics Express, 16:6692–6716

    Article  Google Scholar 

  91. Greengard L, Helsing J (1998) On the numerical evaluation of elastostatic fields in locally isotropic two-dimensional composites. J Mechan Phys of Solids, 46:1441–1462.

    Article  MathSciNet  MATH  Google Scholar 

  92. Greengard L, Moura M (1994) On the numerical evaluation of electrostatic fields in composite materials. Acta Numerica, 3:379–410

    Article  MathSciNet  MATH  Google Scholar 

  93. Greengard L, Rokhlin V (1997) A new version of the fast multipole method for the Laplace equation in three dimensions. Acta Numerica, 6:229–270

    Article  MathSciNet  MATH  Google Scholar 

  94. Gu ST, Wang AL, Xu Y, He QC (2015) Closed-form estimates for the effective conductivity of isotropic composites with spherical particles and general imperfect interfaces. Int. J. Heat Mass Tran. 83, 317–326

    Article  Google Scholar 

  95. Gubernatis IE, Domany E, Krymhansl IA, Huberman M (1977) The Born approximation in the theory of the scattering of elastic waves by flows. J Appl Phys, 48:2812–2819

    Article  Google Scholar 

  96. Gurtin ME, Murdoch AI (1975) A continuum theory of elastic material surfaces.Archive for Rational Mechanics and Analysis, 59:291–323.

    Google Scholar 

  97. Guseva O, Lusti HR, Gusev AA (2004) Matching thermal expension of mica-polymer nanocomposites and metals. Model Simul Mater Sci Engng , 12:S101–S105

    Article  Google Scholar 

  98. Halle DK (1976) The physical properties of composite materials J Mater Sci, 11:2105–2141

    Google Scholar 

  99. Hansen JP, McDonald IR (1986) Theory of Simple Liquids. Academic Press, New York

    MATH  Google Scholar 

  100. Hashin Z (1968) Assessment of the self consistent scheme approximation: conductivity of particulate composites. J Compos Mater, 2:284–300

    Article  Google Scholar 

  101. Hashin Z (1972) Theory of Fiber Reinforced Materials. NASA Contractor report CR-1974, NASA, Washington, DC

    Google Scholar 

  102. Hashin Z (1991a) Thermoelastic properties of particular composites with imperfect interface. J Mech Phys Solids, 39:745–762

    Article  MathSciNet  Google Scholar 

  103. Hashin Z (1991b) The spherical inclusion with imperfect interface. J. Appl. Mech., 58:444–449

    Article  Google Scholar 

  104. Hashin Z (2002) Thin interphase imperfect interface in elasticity with application to coated fiber composites. J. Mech. Phys. Solids., 50:2509–2537

    Article  MathSciNet  MATH  Google Scholar 

  105. Hashin Z, Shtrikman S (1962a) On some variational principles in anisotropic and nonhomogeneous elasticity. J Mech Phys Solids, 10:335–342

    Article  MathSciNet  MATH  Google Scholar 

  106. Hashin Z, Shtrikman S (1962c) A variational approach to the theory of the effective magnetic permeability of multiphase materials. J Appl Phys, 35:3125–3131

    Article  MATH  Google Scholar 

  107. Hatta H, Taya M (1985) Effective thermal conductivity of a misoriented short fiber composite. J Appl Phys, 58:2478–2486

    Article  Google Scholar 

  108. Hatta H, Taya M (1986) Equivalent inclusion method for steady state heat conduction in composites. Int J Engng Sci, 24:1159–1172

    Article  MATH  Google Scholar 

  109. Hill R (1963a) Elastic properties of reinforced solids: some theoretical principles. J Mech Phys Solids, 11:357–372

    Article  MATH  Google Scholar 

  110. Hill R (1963b) New derivations of some elastic extremum principles. Prog in Appl Mechanics. The Prager Anniversary Volume. Macmillan, New York, 99–106

    Google Scholar 

  111. Hill R (1965) A self-consistent mechanics of composite materials. J Mech Phys Solids, 13:212–222

    Article  Google Scholar 

  112. Honein E, Honein T, Herrmann G (1990) On two circular inclusions in harmonic problems. Quart Appl Math, 50:479–499

    Article  MathSciNet  MATH  Google Scholar 

  113. Huang JH, Kuo W-S (1997) The analysis of piezoelectric/piezomagnetic composite materials containing ellipsoidal inclusions J. Applied Physics, 81:1378–1386

    Google Scholar 

  114. Huang JH, Liu H-K, Dai W-L (2000) The optimized fiber volume fraction for magnetoelectric coupling effect in piezoelectric-piezomagnetic continuous fiber reinforced composites. Int J Engng Sci, 38:1207–1217

    Article  Google Scholar 

  115. Hussein MI, Leamy MJ, Ruzzene M (2014) Dynamics of phononic materials and structures: Historical origins, recent progress, and future outlook. Applied Mechanics Reviews, 66:040802 (52 pages)

    Article  Google Scholar 

  116. Jayaraman K, Reifsnider KL (1992) Residual stresses in a composite with continuously varying Young’s modulus in the fiber/matrix interphase. J. Comp. Mater. 26:770–791

    Article  Google Scholar 

  117. Jeffrey DJ (1973) Conduction through a random suspension of spheres. Proc Roy Soc Lond, A335:355–367

    Google Scholar 

  118. Jeffrey DJ (1974) Group expansion for the bulk properties of a statistically homogeneous, random suspension. Proc Roy Soc London, A338:505–516

    MathSciNet  Google Scholar 

  119. Jiang B, Fang D-N, Hwang K-C (1999) A unified model for piezocomposites with non-piezoelectric matrix and piezoelectric inclusions. Int J Solids Structure, 37:2707–2733

    Article  MATH  Google Scholar 

  120. Kanaun SK, Levin VM (1984) Development of effective wave operator for medium with isolated inhomogeneities. Mech Solids, (5):67–76 (In Russian)

    Google Scholar 

  121. Kanaun SK, Levin VM (1994) Effective field method on mechanics of matrix composite materials. In: Markov KZ (ed), Advances in Math Modelling of Composite Materials. World Scientific, Singapore, 1–58

    Google Scholar 

  122. Kanaun KK, Levin VM (2008) Self-Consistent Methods for Composites. Vol. 1, 2, Springer, Dordrecht

    Book  MATH  Google Scholar 

  123. Kanaun SK, Levin VM, Sabina FJ (2004) Propagation of elastic waves in composites with random set of spherical inclusions (effective medium approach). Wave Motion, 40:69–88

    Article  MathSciNet  MATH  Google Scholar 

  124. Keiller RA, Feuillebous F (1993) Head conduction through an inhomogeneous suspension. Proc Roy Soc Lond, A440:717–726

    MATH  Google Scholar 

  125. Keller JB (1963) Conductivity of a medium containing a dense array of perfectly conducting spheres or cylinders or nonconducting cylinders. J Appl Phys, 34:991–993

    Article  MATH  Google Scholar 

  126. Khoroshun LP (1972) Elastic properties of materials reinforced by uni-directional short fibers. Prikladnaya Mekhanika, 8(12), 86–92 (In Russian. Engl. Transl. Soviet Appl. Mech. 8:1358–1363)

    Google Scholar 

  127. Khoroshun LP (1978) Random functions theory in problems on the macroscopic characteristics of microinhomogeneous media. Priklad Mekh, 14(2):3–17 (In Russian. Engl Transl. Soviet Appl Mech, 14:113–124)

    Google Scholar 

  128. Khoroshun LP, Dorodnykh TI (2004) The effective piezoelectric properties of polycrystals with the trigonal symmetry. Acta Mechan, 169:203–219

    Article  MATH  Google Scholar 

  129. Khoroshun LP, Maslov BP, Leshchenko PV (1989) Prediction of Effective Properties of Piezoactive Composites. Naukova Dumka, Kiev (In Russian)

    Google Scholar 

  130. Khoroshun LP, Maslov BP, Shikula EN, Nazarenko LV (1993) Statistical Mechanics and Effective Properties of Materials. Naukova Dumka, Kiev (In Russian)

    Google Scholar 

  131. Kirkpatrick S (1973) Percolation and conductivity. Rev Mod Phys, 45:574–588

    Article  Google Scholar 

  132. Koelman JM, Kuijper A (1997) An effective medium model for the effective conductivity of N-component anisotropic and percolating mixture. Physica, A247:10–22

    Article  Google Scholar 

  133. Kröner E (1958) Berechnung der elastischen Konstanten des Vielkristalls aus den Konstanstanten des Einkristalls. Z Physik, 151:504–518

    Article  Google Scholar 

  134. Kröner E (1967b) Elasticity theory of materials with long range cohesive forces. Int J Solid Struct, 3:731–742

    Article  MATH  Google Scholar 

  135. Kröner E (1977) Bounds for effective moduli of disordered materials. J Mech Phys Solids, 25:137–155

    Article  MATH  Google Scholar 

  136. Kröner E (1990) Modified Green function in the theory of heterogeneous and/or anisotropic linearly elastic media. In: Weng GJ, Taya M, Abe H (eds) Micromechanics and Inhomogeneity. The Toshio Mura 65th Anniversary Volume. Springer–Verlag, New York, 197–211

    Google Scholar 

  137. Kuo WS, Huang JH (1997) On the effective electroelastic properties of piezoelectric composites containing spatially oriented inclusions. Int J Solids Struct, 34:2445-2461

    Article  MATH  Google Scholar 

  138. Kushch (2018) Stress field and effective elastic moduli of periodic spheroidal particle composite with Gurtin–Murdoch interface. Int. J. Eng. Sci., 132, 79–96

    Article  MathSciNet  MATH  Google Scholar 

  139. Kushch V (2020) Micromechanics of Composites Multipole Expansion Approach. Amsterdam Butterworth-Heinemann.

    Google Scholar 

  140. Kushch (2020b) Elastic fields and effective stiffness of ellipsoidal particle composite using the representative unit cell model and multipole expansion method. Int. J. Engineering Science, 154:103336

    Article  MathSciNet  MATH  Google Scholar 

  141. Kushch (2020c) Numerical algorithm of multipole expansion method for conductivity of ellipsoidal particle composite J. Computational Physics, 418: 109642

    Google Scholar 

  142. Kushch VI, Sevostianov I, Giraud A (2017) Local fields and effective conductivity tensor of ellipsoidal particle composite with anisotropic constituents. Proc. R. Soc. A, 473:20170472.

    Article  MathSciNet  MATH  Google Scholar 

  143. Kussow A-G, Akyurtlu A, Angkawisittpan N (2008) Optically isotropic negative index of refraction metamaterial Phys. Stat. Sol., b245:992–997

    Google Scholar 

  144. Kuznetsov S, Fish J (2012) Mathematical homogenization theory for electroactive continuum. Int. J. Numer. Meth. Engng, 91:1199–1226

    Article  MathSciNet  Google Scholar 

  145. Landau LD, Lifshitz EM (1960) Electrodynamics of Continuum Media. Pergamon Press, Oxford

    MATH  Google Scholar 

  146. Landauer R (1952) The electrical resistance of binary metallic mixtures. J Appl Phys, 23:779–784

    Article  Google Scholar 

  147. Landauer R (1978) Electric conductivity in inhomogeneous media. In: Garland JC, Tanner DB (eds) Electric, Transport and Optical Properties of Inhomogeneous Media. American Institute of Physics, New York, 2–43

    Google Scholar 

  148. Lax M (1952) Multiple scattering of waves II. The effective fields dense systems. Phys Rev, 85:621–629

    Article  MATH  Google Scholar 

  149. Lee JH, Singer JP, Thomas EL (2012) Micro-/nanostructured mechanical metamaterials Adv. Mater., 24:4782–4810

    Google Scholar 

  150. Lekhnitskii AG (1963) Theory of Elasticity of an Anisotropic Elastic Body. Holder Day, San Francisco

    MATH  Google Scholar 

  151. Levi O, Bergman DJ (1994) Critical behavior of the weakly nonlinear conductivity and flicker noise of two-component composites. Phys Rev, B50:3652–3660

    Article  Google Scholar 

  152. Levin VM (1967) Thermal expansion coefficient of heterogeneous materials. Izv AN SSSR, Mekh Tverd Tela, (2):88–94 (In Russian. Engl Transl. Mech Solids, 2(2):58–61)

    Google Scholar 

  153. Levin VM, Michelitsch TM, Gao H (2002) Propagation of electroacoustic waves in the transversely isotropic piezoelectric medium reinforced by randomly distributed cylindrical inhomogeneities. Int. J. Solids and Structures, 39:5013–5051

    Article  MATH  Google Scholar 

  154. Levin VM, Rakovskaja M. I, Kreher W. S (1999) The effective thermoelectroelastic properties of microinhomogeneous materials. Int J Solids Struct, 36:2683–2705

    Article  MATH  Google Scholar 

  155. Levin VM, Valdiviezo-Mijangos O, Sabina GJ (2011) Propagation of electroacoustic axial shear waves in a piezoelectric medium reinforced by continuous fibers. Int. J. Engng Science, 49:1232–1243.

    Article  MathSciNet  MATH  Google Scholar 

  156. Li JY (2000a) The effective electroelastic moduli of textured piezoelectric polycrystalline aggregates. J Mech Phys Solids, 48:529–552

    Article  MATH  Google Scholar 

  157. Li JY (2000b) Magnetoelectroelastic multi-inclusion and inhomogeneity problems and their applications in composite materials. Int J Engng Sci, 38:1993–2011

    Article  Google Scholar 

  158. Li JY (2004) The effective pyroelectric and thermal expansion coefficients of ferroelectric ceramics. Mechan Mater 36:949-958

    Article  Google Scholar 

  159. Liu B, Zhao X, Zhu W, Luo W, Cheng X (2008) Multiple pass-band optical left-Handed metamaterials based on random dendritic cells. Adv. Funct. Mater., 18:3523–3528

    Article  Google Scholar 

  160. Liu YJ, Mukherjee S, Nishimura N, Schanz M, Ye W, Sutradhar A, Pan E, Dumont NA, Frangi A, Saez A (2011) Recent advances and emerging applications of the boundary element method. Applied Mechanics Reviews, 64:031001 (38 pages)

    Article  Google Scholar 

  161. Lu SY, Lin HC (1996a) Effective conductivity of composites with spherical inclusions: effect of coating and detachment J Appl Phys, 79:609–618

    Google Scholar 

  162. Lu SY, Lin HC (1996b) Effective conductivity of composites containing aligned spheroidal inclusions of finite conductivity J Appl Phys, 79:6761–6769

    Google Scholar 

  163. Lu XY, Li H, Wang B, (2011) Theoretical analysis of electric, magnetic and magnetoelectric properties of nano-structured multiferroic composites. J. Mech. Phys. Solids, 59:1966–1977

    Article  MathSciNet  MATH  Google Scholar 

  164. Mal AK, Knopoff L (1967) Elastic wave velocities in two-component systems. I. Inst Math Appl, 3:376–387

    Article  MATH  Google Scholar 

  165. Marchioro M, Prosperitti A (1999) Heat conduction in a non-uniform composite with spherical inclusions. Proc Roy Soc Lond, A455:1483–1508

    Article  MathSciNet  MATH  Google Scholar 

  166. Markov KZ (2000) Elementary micromechanics of heterogeneous media. In: Markov K, Preziosi L (eds), Heterogeneous Media. Micromechanics, Modelling, Methods, and Simulations. Birkhäuser, Boston, 1–162

    Google Scholar 

  167. Maugin GA (1988) Continuum Mechanics of Electromagnetic Solids. North-Holland, Amsterdam

    MATH  Google Scholar 

  168. McKenzie DR, McPedran RC, Derrick GH (1978) The conductivity of lattices of spheres. II. The body centered and face centered cubic lattices. Proc Roy Soc Lond, A362:211–232

    Google Scholar 

  169. McLaughlin R (1977) A study of the differential scheme for composite materials. Int J Engrg Sc, 15:237–244

    Article  MATH  Google Scholar 

  170. McPedran RC, McKenzie DR (1978) The conductivity of lattices of spheres. I. The simple cubic lattice. Proc Roy Soc Lond, A359:45–63

    Google Scholar 

  171. Meridith RE, Tobias CW (1960) Resistance to potential flow through a cubical array of spheres. J Appl Phys, 31:1270–1274

    Article  Google Scholar 

  172. Milgrom M (1990) Linear response of general composite systems to many coupled fields. Phys. Rev. B 41:12484–12494

    Google Scholar 

  173. Milgrom M, Shtrikman S (1989) Linear response of two-phase composites with cross moduli: exact universal relations. Phys Rev, A40:1568–1575

    Article  Google Scholar 

  174. Miloh T, Benveniste Y (1988) A generalized self-consistent method for the effective conductivity of composites with ellipsoidal inclusions and cracked bodies J Appl Phys, 63:789–7796

    Google Scholar 

  175. Miloh T, Benveniste Y (1999) On the effective conductivity of composites with ellipsoidal inhomogeneities and highly conducting interfaces. Proc Roy Soc Lond, A455:2687–2706

    Article  MathSciNet  MATH  Google Scholar 

  176. Milton GW (1981) Bounds on the transport and optical properties of a two-component composite material. J Appl Phys, 52:5294–5304

    Article  Google Scholar 

  177. Milton GW (1982) Bounds on the elastic and transport properties of two-component composites. J Mech Phys Solids, 30:177–191

    Article  MathSciNet  MATH  Google Scholar 

  178. Milton GW (1985) The coherent potential approximation is a realizable effective medium scheme. Commun Math Phys, 99:463–500

    Article  MathSciNet  Google Scholar 

  179. Milton GW (2002) The Theory of Composites. Cambridge University Press, Cambridge, UK

    Book  MATH  Google Scholar 

  180. Mogilevskaya SG, Kushch VI, Zemlyanova AY (2019) Displacements representations for the problems with spherical and circular material surfaces. The Quarterly J. Mechanics App. Mathematics, 72, 449–471

    Article  MathSciNet  MATH  Google Scholar 

  181. Mori T, Tanaka K (1973) Average stress in matrix and average elastic energy of materials with misfitting inclusions. Acta Metall, 21:571–574

    Article  Google Scholar 

  182. Moschovidis ZA, Mura T (1975) Two-ellipsoidal inhomogeneities by the equivalent inclusion method. J Appl Mech, 42:847–852

    Article  MATH  Google Scholar 

  183. Mura T (1987) Micromechanics of Defects in Solids. Martinus Nijhoff, Dordrecht

    Book  MATH  Google Scholar 

  184. Nan C-W, Clarke DR (1997) Effective properties of ferroelectric and/or ferromagnetic composites: a unified approach and its application. J Am Ceram Soc, 60:1333–1340

    Google Scholar 

  185. Nemat-Nasser S, Hori M (1993) Micromechanics: Overall Properties of Heterogeneous Materials. Elsevier, North-Holland.

    MATH  Google Scholar 

  186. Noh TW, Song PH, Sievers AJ (1991) Self-consistency conditions for the effective medium approximation in composite materials. Phys Rev, B44:5459–5464

    Article  Google Scholar 

  187. Norris AN, Callegari AJ, Sheng PA (1985) A generalized differential effective medium theory. J Mech Phys Solids, 33:525–543

    Article  MATH  Google Scholar 

  188. O’Connel RJ, Budiansky B (1974) Seismic velocities in dry and saturated cracked solids. J Geophys Res, 79:5412–5426

    Article  Google Scholar 

  189. Parton VZ, Kudryavtsev BA (1988) Electromagnetic Elasticity of Piezoelectric and Electrically Conductive Bodies. Nauka, Moscow

    Google Scholar 

  190. Pavanello F, Manca F, Palla PL, Giordano S (2012) Generalized interface models for transport phenomena Unusual scale effects in composite nanomaterials. J. Appl. Phys., 112,084306

    Article  Google Scholar 

  191. Phan-Thien N, Pham DC (2000) Differential multiphase models for polydispersed spheroidal inclusions: thermal conductivity and effective viscosity. Int J Engng Sci, 38:73–88

    Article  Google Scholar 

  192. Phelan PE, Niemann RC (1998) Effective thermal conductivity of a thin, randomly oriented composite material. J Heat Transfer, 120:971–976

    Article  Google Scholar 

  193. Pfeil K, Klingenberga DJ (2004) Nonlocal electrostatics in heterogeneous suspensions using a point-dipole model J. Appl. Phys., 96:5341–5348

    Google Scholar 

  194. Plankensteiner AF, Böhm HJ, Rammerstorfer FG, Buryachenko VA (1996) Hierarchical modeling of the mechanical behavior of high speed steels as layer–structured particulate MMCs. J Physique IV, 6:C6-395–C6-402

    Google Scholar 

  195. Polder D, Van Santen JH (1946) The effective permeability of mixtures of solids. Physica, XII:257–271

    Google Scholar 

  196. Polizzotto C (2003) Gradient elasticity and nonstandard boundary conditions. Int. J. Solids Structures, 40:7399–7423

    Article  MathSciNet  MATH  Google Scholar 

  197. Ponte Castañeda P, Willis JR (1995) The effect of spatial distribution on the effective behavior of composite materials and cracked media. J Mech Phys Solids, 43:1919–1951

    Article  MathSciNet  MATH  Google Scholar 

  198. Qin Q, Yang Q-S (2008) Macro-Micro Theory on Multifield Coupling Behavior of Heterogeneous Materials. Springer, Berlin

    Google Scholar 

  199. Qu J (1993) The effect of slightly weakened interfaces on the overall elastic properties of composite materials. Mech Mater, 14:269–281

    Article  Google Scholar 

  200. Rogula D (1982) Nonlocal theory of material media. CISM Courses and Lectures, 268. Springer-Verlag, Vienna, New York

    Google Scholar 

  201. Sabina FJ, Smyshlyaev VP, Willis JR (1993) Self-consistent analysis of waves in a matrix-inclusion composite.–I. Aligned spheroidal inclusions. J Mech Phys Solids, 41:1573–1588

    Article  MathSciNet  MATH  Google Scholar 

  202. Sahimi M (1998) Non-linear and non-local transport processes in heterogeneous media: from long-range correlated percolation to fracture and materials breakdown. Phys Rep, 306:213–395

    Article  MathSciNet  Google Scholar 

  203. Sahimi M (2003) Heterogeneous Materials II Nonlinear and Breakdown Properties, Springer-Verlag, Berlin

    MATH  Google Scholar 

  204. Sangani AS, Yao C (1997) Transport processes in random array of cylinders. I. Thermal conduction. Phys Fluids, 31:2426–2434

    Article  MATH  Google Scholar 

  205. Sarychev AK, Shalaev VM (20000) Electromagnetic field fluctuations and optical nonlinearities in metal-dielectric composites. Physics Reports, 335:275–371

    Google Scholar 

  206. Sato H, Shindo Y (2002) Influence of microstructure on scattering of plane elastic waves by a distribution of partially debonded elliptical inclusions. Mech Mater 34:401–409

    Article  Google Scholar 

  207. Schulgasser K (1976a) Relationship between single-crystal and polycrystal electrical conductivity. J Appl Phys, 47:1880–1886

    Article  Google Scholar 

  208. Schulgasser K (1976b) On the conductivity of fiber-reinforced materials. J Math Phys, 17:382–387

    Article  Google Scholar 

  209. Sejnoha M, Zeman J (2013) Micromechanics in Practice. WIT Press, Southampton, UK

    Google Scholar 

  210. Sen AK, Torquato S (1989) Effective conductivity of anisotropic two-phase composite medium. Phys Rev, B39:4504–4515

    Article  Google Scholar 

  211. Sen P, Scala C, Cohen MH (1981) A self-similar model for sedimentary rocks with application to the dielectric constant of fused glass beards. Geophysics, 46:781–795

    Article  Google Scholar 

  212. Serpilli M, Rizzoni R, Lebon F, Dumont S (2019). An asymptotic derivation of a general imperfect interface law for linear multiphysics composites. Int. J. Solids Struct., 180–181: 97–107

    Article  Google Scholar 

  213. Sharma P, Ganti S (2004) Size-dependent Eshelby-s tensor for embedded nano-inclusions incorporating surface/interface energies. J Appl Mech, 71:663–671

    Article  MATH  Google Scholar 

  214. Sharma, P. and Wheeler, L.T, (2007) Size-dependent elastic state of ellipsoidal nano-inclusions incorporating surface/interface tension. J. Applied Mechanics, 74:447–454

    Article  MathSciNet  MATH  Google Scholar 

  215. Shermergor TD (1977) The Theory of Elasticity of Microinhomogeneous Media. Nauka, Moscow (In Russian)

    Google Scholar 

  216. Shermergor TD, Yakovlev VB (1993) Concentration of coupled electrical mechanical fields on a crystallite surface in textured quartz. Izv Acad Sci Russ Phys Solid Earth, 32:89–94 (In Russian)

    Google Scholar 

  217. Shklovskii BI, Efros AL (1979) Electronic Properties of Doped Semiconductors. Nauka, Moscow (In Russian)

    Google Scholar 

  218. Shvidler MI (1985) Statistical Hydrodynamics of Porous Media. Nauka, Moscow. (In Russian)

    MATH  Google Scholar 

  219. Stagfen ESG (1988) A nonlocal theory for the heat transport in composites containing highly conducting fibrous inclusions. Phys Fluids, 31:2405–2425

    Article  MATH  Google Scholar 

  220. Steigmann DJ, Ogden RW (1999) Elastic surface-substrate interactions. Proc. R. Soc. Lond., A455, 437–474

    Article  MathSciNet  MATH  Google Scholar 

  221. Stoy RD (1989) Solution procedure for the Laplace equation in bispherical coordinates for two sphere in uniform external field: parallel orientation. J Appl Phys, 65:2611–2615

    Article  Google Scholar 

  222. Straley JP (1981) Thermoelectric properties of inhomogeneous materials. J Phys D: Appl Phys, 14:2101–2105

    Article  Google Scholar 

  223. Stratonovich RL (1963) Topics in the Theory of Random Noise. Gordon and Breach, New York

    Google Scholar 

  224. Stroud D (1998) The effective medium approximation: some recent development. Superlatt Microstruct, 23:567–573

    Article  Google Scholar 

  225. Stroud D, Hui PM (1988) Nonlinear succeptibilities of granular materials. Phys Rev, B37:8719–8724

    Article  Google Scholar 

  226. Talbot DR, Willis JR (1982a) Variational estimates for dispersion and attenuation of waves in random composites. I. General theory. Int I Solids Struct, 18:673–683

    Article  MATH  Google Scholar 

  227. Talbot DRS, Willis JR (1982b) Variational estimates for dispersion and attenuation of waves in random composites. II. Isotropic composites. Int I Solids Struct, 18:685–698

    Article  MATH  Google Scholar 

  228. Tan H, Huang Y, Liu C, Inglis HM, Ravichandran G, Geubelle PH (2007) The uniaxial tension of particle-reinforced composite materials with nonlinear interface debonding. Int. J. Solids Struct. 44:1809–1822

    Article  MATH  Google Scholar 

  229. Tan H, Huang Y, Liu C, Ravichandran G, Paulino GH (2007) Constitutive behaviors of composites with interface debonding the extended Mori–Tanaka method for uniaxial tension. Int. J. Fract., 146:139–148

    Article  MATH  Google Scholar 

  230. Thorpe MF (1992) The conductivity of a sheet containing a few polygonal holes and/or superconducting inclusions Proc Roy Soc Lond, A437:215–227

    Google Scholar 

  231. Torquato S (1980) Microscopic Approach to Transport in Two-Phase Random Media. PhD Thesis, State University of New York at Stony Brook

    Google Scholar 

  232. Torquato S (2002a) Random Heterogeneous Materials: Microstucture and Macroscopic Properties. Springer-Verlag, New York, Berlin

    Book  MATH  Google Scholar 

  233. Torquato S, Lado F (1992) Improved bounds on the effective elastic moduli of random arrays of cylinders. J Appl Mech, 59:1–6

    Article  MathSciNet  Google Scholar 

  234. Torquato S, Rintoul MD (1995) Effect of the interface on the properties of composite media. Phys Rev Lett, 75:4067–4070

    Article  Google Scholar 

  235. Varadan VV, Kim K (2012) Fabrication of 3-D metamaterials using LTCC techniques for high-frequency applications IEEE Trans. Components, Packaging Manufac. Technol., 2:410–417

    Google Scholar 

  236. Varadan VK, Ma Y, Varadan VV (1985) A multiple scattering theory for elastic wave propagation in discrete random media. J Acoust Soc Amer, 77:375–385

    Article  MATH  Google Scholar 

  237. Wang Z, Zhu J, ** XY, Chen WQ, Zhan C (2014) Effective moduli of ellipsoidal particle reinforced piezoelectric composites with imperfect interfaces J. Mechanics Physics Solids, 65: 138–156

    Google Scholar 

  238. Wani SN, Sangani AS, Sureshkumar R (2012) Effective permittivity of dense random particulate plasmonic composites J. Opt. Soc. Am., B 29:1443–1455.

    Google Scholar 

  239. Weber L, Fischer C, Mortensen A (2003) On the influence of the shape of randomly oriented, non-conducting inclusions in a conducting matrix on the effective electrical conductivity. Acta Mater, 51:495–505

    Article  Google Scholar 

  240. Weng GJ (1990) The theoretical connection between Mori–Tanaka’s theory and the Hashin–Shtrikman–Walpole bounds. Int J Engng Sci, 28:1111–1120

    Article  MathSciNet  MATH  Google Scholar 

  241. Willis JR (1977) Variational and related methods for the overall properties and self-consistent estimates for the overall properties. J Mech Phys Solids, 25:185–202

    Article  MATH  Google Scholar 

  242. Willis JR (1978) Variational principles and bounds for the overall properties of composites. In: Provan JW (ed), Continuum Models of Disordered Systems. University of Waterloo Press, Waterloo 185–215

    Google Scholar 

  243. Willis JR (1980a) A polarization approach to the scattering of elastic waves I. Scattering by a single inclusion. J Mech Phys Solids, 28:287–305

    Article  MathSciNet  MATH  Google Scholar 

  244. Willis JR (1980b) A polarization approach to the scattering of elastic waves. II: Multiple scattering from inclusions. J Mech Phys Solids, 28:307–326

    Article  MathSciNet  MATH  Google Scholar 

  245. Willis JR (1981a) Variational and related methods for the overall properties of composites. Adv Appl Mech. 21:1–78

    Article  MathSciNet  MATH  Google Scholar 

  246. Willis IR (1981b) Variational principles for dynamic problems for inhomogeneous elastic media. Wave Motion, 3:1–11

    Article  MathSciNet  MATH  Google Scholar 

  247. Willis JR (1997) Dynamics of composites. In Continuum Micromechanics, CISM Lecture Notes. Springer, New York, NY, 265–290

    Google Scholar 

  248. Wu RS, Aki K (1985) Elastic wave scattering by a random medium and small-scale inhomogeneities in the lithosphere. J Geoph Res, B90:10261–10273

    Article  Google Scholar 

  249. Yin HM, Paulino GH, Buttlar WG, Sun LZ (2005) Effective thermal conductivity of two-phase functionally graded particulate composites. J Appl Phys, 98:063704

    Article  Google Scholar 

  250. Yin HM, Paulino GH, Buttlar WG, Sun LZ (2007) Micromechanics-based thermoelastic model for functionally graded particulate materials with particle interactions. J Mech Phys Solids, 55:132–160

    Article  MATH  Google Scholar 

  251. You LH, You XY, Zheng ZY (2006) Thermomechanical analysis of elastic–plastic fibrous composites comprising an inhomogeneous interphase Computational Materials Science, 36:440–450.

    Google Scholar 

  252. Zabihyan R, Mergheima J, Javili A, Steinmanna P (2018) Aspects of computational homogenization in magneto-mechanics Boundary conditions, RVE size and microstructure composition Int. J. Solids Structures, 130–131:105–121

    Google Scholar 

  253. Zeller R, Dederichs PH (1973) Elastic constants of polycrystals. Phys. Stat. Sol., a55:831–842

    Google Scholar 

  254. Zeng X, Bergman DJ, Hui PM, Stroud D (1988) Effective medium theory for weakly nonlinear composites. Phys Rev, B38:10970–10973

    Article  Google Scholar 

  255. Ziman JM (1979) Models of Disorder. Cambridge University Press, New York

    Google Scholar 

  256. Zimmerman RW (1996) Effective conductivity of a two-dimensional medium containing elliptical inhomogeneities Proc Roy Soc Lond, A452:1713–1727

    Google Scholar 

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Buryachenko, V.A. (2022). Subsequent Generalizations of Theory and Related Problems. In: Local and Nonlocal Micromechanics of Heterogeneous Materials. Springer, Cham. https://doi.org/10.1007/978-3-030-81784-8_15

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