In this paper, linearly thermoelastic composite media are treated, which consist of a homogeneous matrix containing a statistically homogeneous random set of ellipsoidal uncoated or coated inclusions. Effective properties (such as compliance, thermal expansion, stored energy) as well as the first statistical moments of stresses in the components are estimated for both the general case of nonhomogeneity of the thermoelastic inclusion properties and arbitrary choice of comparison medium. The micromechanical approach is based on Green's function techniques as well as on the generalization of the "multiparticle effective field" method (MEFM), previously proposed for the estimation of stress field averages in the components for the case of coinciding elastic moduli of the matrix and comparison medium. The author considers in detail the connection of methods proposed with numerous related methods and demonstrates that the MEFM includes, as particular cases, the well-known methods of mechanics of strongly heterogeneous media (such as the effective medium and the mean field methods and some others).
[1] Willis, J. R.: The overall elastic response of composite materials. ASME J. Appl. Mech., 50, 1202-1209 (1983).
2.
[2] Shermergor, T D.: The Theory of Elasticity of Microinhomogeneous Media, Nauka, Moscow, 1977 (in Russian).
3.
[3] Christensen, R. M.: Mechanics of Composite Materials, Wiley Interscience, New York, 1979.
4.
[4] Willis, J. R.: Variational and related methods for the overall properties of composites. Advances in Applied Mechanics, 21, 1-78 (1981).
5.
[5] Kanaun, S. K.: Elastic medium with random fields of inhomogeneities. Elastic Media with Microstructure, ed. I. A. Kunin, vol. 2, pp. 165-228, Springer-Verlag, Berlin, 1983.
6.
[6] Mura, T.: Micromechanics of Defects in Solids, Martinus Nijhoff, Dordrecht, 1987.
7.
[7] Kreher, W. and Pompe, W.: Internal Stresses in Heterogeneous Solids, Akademie-Verlag, Berlin, 1989.
8.
[8] Nemat-Nasser, S. and Hori, M.: Micromechanics: Overall Properties of Heterogeneous Materials, Elsevier, North Holland, 1993.
9.
[9] Kanaun, S. K. and Levin, V M.: Effective field method in mechanics of matrix composite materials. Advances in Mathematical Modeling of Composite Materials, ed. K. Z. Markov, pp. 1-58, World Scientific, Singapore, 1994.
10.
[10] Markov, K. Z.: Elementary micromechanics of heterogeneous media. Heterogeneous Media: Micromechanics, Modeling, Methods, and Simulations, ed. K. Markov and L. Preziosi, pp. 1-162, Birkhluser, Boston, 1999.
11.
[11] Khoroshun, L. P.: Mathematical models and methods of the mechanics of stochastic composites. Prikl. Mekh., 30, 30-62 (2000) (in Russian, Engl. Transl. Int. Appl. Mech. 30, 1284-1316.).
12.
[12] Buryachenko, V A.: Multiparticle effective field and related methods in micromechanics of composite materials. Appl. Mech. Reviews, 54, 1-47 (2001).
13.
[13] Krdner, E.: Berechnung der elastischen Konstanten des Vielkristalls aus den Konstanstanten des Einkristalls. Z. Physik., 151, 504-518 (1958).
14.
[14] Hill, R.: A self-consistent mechanics of composite materials. J. Mech. Phys. Solids, 13, 212-222 (1965).
15.
[15] Mori, T. and Tanaka, K.: Average stress in matrix and average elastic energy of materials with misfitting inclusions. Acta Metall, 21, 571-574 (1973).
16.
[16] Benveniste, Y.: A new approach to application of Mori-Tanaka's theory in composite materials. Mech. Mater, 6, 147-157 (1987).
17.
[17] Morse, P M. and Feshbach, H.: Methods of Theoretical Physics: Parts I and II, McGraw-Hill, New York, 1953.
18.
[18] Foldy, L. L.: The multiple scattering of waves: 1. General theory of isotropic scatering by randomly distributed scatters. Physical Review, 67, 107-117 (1945).
19.
[19] Lax, M.: Multiple scattering of waves: II. The effective fields dense systems. Phys. Rev, 85, 621-629 (1952).
20.
[20] Chaban, I. A.: Self-consistent field approach to calculation of the effective parameters of microinhomogeneous media. Akust. Zhurn., 10, 351-358 (1965) (in Russian, Engl. Transl. Soviet Physics-Acoustics 10, 298-304.).
21.
[21] Kachanov, M.: Elastic solids with many cracks and related problems. Advances in Applied Mechanics, 30, 259-445 (1993).
22.
[22] Buryachenko, V A. and Kreher, W S.: Internal residual stresses in heterogeneous solids: A statistical theory for particulate composites. J. Mech. Phys. Solids, 43, 1105-1125 (1995).
23.
[23] Buryachenko, V A. and Rammerstorfer, E G.: Elastic stress fluctuations in random structure particulate composites. Europ. J. Mech. A/Solids, 16, 79-102 (1997).
24.
[24] Lekhnitskii, A. G.: Theory of Elasticity of an Anisotropic Elastic Body, Holder Day, San Francisco, 1963.
25.
[25] Buryachenko, V A.: Locality principle and general integral equations of micromechanics of matrix composite materials. Mathematics and Mechanics of Solids, 6, 299-321 (2001).
26.
[26] Gel'fand, I. A. and Shilov, G.: Generalized Functions, vol. 1, Academic Press, New York, 1964.
27.
[27] Buryachenko, V A. and Parton, V Z.: One-particle approximation of the effective field method in the statics of composites. Mekh. Kompoz. Mater, 3, 420-425 (1990) (in Russian, Engl. Transl. Mech. Comps. Mater 26(3), 304-309.)
28.
[28] Buryachenko, V A. and Rammerstorfer, F G.: Thermoelastic stress fluctuations in random structure coated particulate composites. European Journal of Mechanics A/Solids, 17, 763-788 (1998).
29.
[29] Buryachenko, V A. and Rammerstorfer, F G.: On the thermostatics of composites with coated inclusions. International Journal of Solids and Structures, 37, 3177-3200 (2000).
30.
[30] Buryachenko, V A. and Shermergor, T D.: Material and field characteristics of piezoelectric rocks: Some exact results. Fiz. Zemli, 8, 32-42 (1995) (in Russian, Engl. Transl. Physics of the Solid Earth 1996 31, 665-674.).
31.
[31] Willis, J. R. and Acton, J. R.: The overall elastic moduli of a dilute suspension of spheres. Q. J. Mech. App L. Mech., 29, 163-177 (1976).
32.
[32] Ju, J. W. and Sun, L. Z.: A novel formulation for the exterior point Eshelby's tensor of an ellipsoidal inclusion. ASME J. Appl. Mech., 66, 570-574 (1999).
33.
[33] Eshelby, J. D.: Elastic inclusion and inhomogeneities. Progress in Solid Mechanics, ed. I. N. Sneddon and R. Hill, vol. 2, pp. 89-140, North-Holland, Amsterdam, 1961.
34.
[34] Dvorak, G. J. and Benveniste, Y.: On transformation strains and uniform fields in multiphace elastic media. Proc. Roy. Soc. London, A437, 291-310 (1992).
35.
[35] Buryachenko, V A.: A simple method of analysis of multiple inclusion interaction problems. Int. J. Comput. Civil & Structural Engng., (2001, in press).
36.
[36] Buryachenko, V A. and Rammerstorfer, F G.: On the thermoelasticity of random structure particulate composites. Z. Angew. Math. Phys., 50, 934-947 (1999).
37.
[37] Buryachenko, V A. and Lipanov, A. M.: Stress concentration at ellipsoidal inclusions and effective thermoelastic properties of composite materials. Priklad. Mekh., 11, 105-111 (1986) (in Russian, Engl. Transl. Soviet Appl. Mech. 22(11), 1103-1109.).
38.
[38] Kunin, I. A.: Elastic Media with Microstructure, vol. 2, Springer-Verlag, Berlin, 1983.
39.
[39] Khoroshun, L. P.: Random functions theory in problems on the macroscopic characteristics of microinhomogeneous media. Priklad. Mekh., 14(2), 3-17 (1978) (in Russian, Engl. Transl. Soviet Appl. Mech. 14, 113-124.).
40.
[40] Khoroshun, L. R?: Conditional-moment method in problems of the mechanics of composites. Priklad Mekh., 23(10), 100-108 (1987) (in Russian, Engl. Transl. Soviet Appl. Mech. 23, 989-998.).
41.
[41] Buryachenko, V A.: Mechanics of random structure composites. Proc. of 9th Int. Conf. on Composites, ed. A. Mizavete, vol. 3, pp. 398-405, Univ. Zaragoza, Spain, 1993.
42.
[42] Buryachenko, V A. and Lipanov, A. M.: Effective field method in the theory of perfect plasticity of composite materials. Priklad. Mekh. Tekhn. Fiz., 3, 149-155 (1989) (in Russian, Engl. Transl. J Appl. Mech. Tech. Phys. 30, 482-487.).
43.
[43] Kanaun, S. K.: Self-consistent averaging schemes in the mechanics of matrix composite materials. Mekhanika Kompozitnikh Materialov, 26, 702-711 (1990) (in Russian, Engl. Transl. Mech. Comps. Mater 26, 984-994.).
44.
[44] Chen, H. S. and Acrivos, A.: The effective elastic moduli of composite materials containing spherical inclusions at non-dilute concentrations. Int. J. Solids Structures, 14, 349-364 (1978).
45.
[45] Buryachenko, V A. and Rammerstorfer, F G.: Micromechanics and nonlocal effects in graded random structure matrix composites. IUTAMSymp. on Transformation Problems in Composite and Active Materials, ed. Y A. Bahei-El-Din and G. J. Dvorak, pp. 197-206, Kluwer Academic, Dordrecht, 1998.
46.
[46] Ponte Castanieda, IP and Willis, J. R.: The effect of spatial distribution on the effective behavior of composite materials and cracked media. J Mech. Phys. Solids, 43, 1919-1951 (1995).
47.
[47] Levin, V M.: Determination of effective elastic moduli of composite materials. Docl. Akad. Nauk SSSR, 220, 1042-1045 (1975) (in Russian, Engl. Transl. Soviet Phys. Docl. 20, 147-148.).
48.
[48] Levin, V M.: On the stress concentration in inclusions in composite materials. Prikl. Matem. Mekh., 41, 735-743 (1977) (in Russian, Engl. Transl. Applied Mathem. Mech. 41,735-743.).
49.
[49] Stang, H.: Strength of composite materials with small cracks in the matrix. Int. J. Solids Structures, 22, 1259-1277 (1986).
50.
[50] Levin, V M.: Thermal expansion coefficient of heterogeneous materials. Izv ANSSSR, Mekh. Tverd. Tela, 2, 88-94 (1967) (in Russian, Engl. Transl. Mech. Solids 2(2), 58-61.).
51.
[51] Rosen, B. W. and Hashin, Z.: Effective thermal expansion coefficient and specific heats of composite materials. Int. J. Engng. Sci., 8, 157-173 (1970).
52.
[52] Buryachenko, V A.: Triply periodical particulate matrix composites in varying external stress fields. Int. J. Solids Structures, 36, 3837-3859 (1999).
53.
[53] Khoroshun, L. P.: On a mathematical model for inhomogeneous deformation of composites. Priklad. Mekh., 32(5), 22-29 (1996) (in Russian, Engl. Transl. Int. Appl. Mech. 32(5), 341-348.).
54.
[54] Willis, J. R.: Bounds and self-consistent estimates for the overall properties of anisotropic composites. J. Mech. Phys. Solids, 25, 185-203 (1977).
55.
[55] Buryachenko, V A. and Bechel, V T.: A series solution of the volume integral equation for multiple inclusion interaction problems. Compos. Sci. Technol., 60, 2465-2469 (2000).
56.
[56] Buryachenko, V A. and Rammerstorfer, F G.: Local effective thermoelastic properties of graded random structure composites. Archive Appl. Mech., 71, 249-272 (2001).
57.
[57] Kreger, I. W.: Rheology of monodisperse lattices. Advances in Colloid and Interface Sci., 3, 111-136 (1972).
58.
[58] Norris, A. N.: An examination of the Mori-Tanaka effective medium approximation for multiphase composites. ASME J. Appl. Mech., 56, 83-88 (1989).
59.
[59] Ferrari, M.: Asymmetry and the high concentration limit of the Mori-Tanaka effective medium theory. Mech. Mater, 11, 251-256 (1991).
60.
[60] Benveniste, Y., Dvorak, G. J., and Chen, T.: On diagonal and elastic symmetry of the approximate effective stiffness tensor of heterogeneous media. J. Mech. Phys. Solids, 39, 929-946 (1991).
61.
[61] Weng, G. J.: The theoretical connection between Mori-Tanaka's theory and the Hashin-Shtrikman-Walpole bounds. Int. J Engng. Sci., 28, 1111-1120 (1990).
62.
[62] Ju, J. W. and Chen, T M.: Micromechanics and effective moduli of elastic composites with randomly dispersed inhomogeneities. Macroscopic Behavior of Heterogeneous Materialsfi-om the Microstructure, pp. 95-109, ASME, ADM 147, New York, 1992.
63.
[63] Ju, J. W. and Chen, T. M.: Effective elastic moduli of two-dimensional brittle solids with interacting microcracks: I. Basic formulations. J. Appl. Mech., 61, 349-357 (1994).
64.
[64] Ju, J. W. and Chen, T M.: Micromechanics and effective moduli of elastic composites containing randomly dispersed ellipsoidal inhomogeneities. Acta Mech., 103, 103-121 (1994).
65.
[65] Ju, J. W. and Chen, T M.: Effective elastic moduli of two-phase composites containing randomly dispersed spherical inhomogeneities. Acta Mech., 103, 123-144 (1994).
66.
[66] Ju, J. W. and Zhang, X. D.: Micromechanics and effective transverse elastic moduli of composites with randomly located aligned circular fibers. Int. J. Solids Structures, 35, 941-960 (1998).
67.
[67] Ju, J. W. and Tseng, K. H.: Effective elastoplastic behavior of two-phase ductile matrix composites: A micromechanical framework. Int. J. Solids Structures, 33, 4327-4291 (1996).
68.
[68] Ju, J. W. and Zhang, X. D.: Effective elastoplastic behaviour of ductile matrix composites containing randomly located aligned circular fibers. Int. J. Solids Structures, 38, 4045-4069 (2001).
69.
[69] Kosheleva, A. A.: Method of multipolar expansion in the mechanics of matrix composites. Mekhanika Kompozititnykh Materialov, 19, 416-422 (1983) (in Russian, Engl. Transl. Mech. Composite Materials 19, 301-307.).
70.
[70] Kachanov, M.: Elastic solids with many cracks: A simple method of analysis. Int. J. Solids Structures, 23, 23-43 (1987).
71.
[71] Zhou, S. A. and Hsieh, R. K.: Statistical theory of elastic materials with micro-defects. Int. J. Engng Sci., 24, 1195-1206 (1986).
72.
[72] Fassi-Fehri, O., Hihi, A., and Berveiller, M.: Multiple site self consistent scheme. Int. J. Engng. Sci., 27, 495-502 (1989).
73.
[73] Lipinski, P. and Berveiller, M.: Elastoplasticity of micro-inhomogeneous metals at large strains. Int. J. Plasticity, 5, 149-172 (1989).
74.
[74] Lipinski, P., Berveiller, M., Reubrez, E., and Morreale, J.: Transition theories of elastic-plastic deformation of metallic polycrystals. Archive of Appl. Mechanics, 65, 295-311 (1995).
75.
[75] Kanaun, S. K.: Self-consistent field approximation for an elastic composite medium. Zhurnal Prikladnoi i Tehknich. Fiziki, 18(2), 160-169 (1977) (in Russian, Engl. Transl. J. Applied Mech. Techn. Physics, 18, 274-282.).
76.
[76] Ju, J. W. and Tseng, K. H.: A three-dimensional micromechanical theory for brittle solids with interacting microcracks. Int. J Damage Mechanics, 1, 102-131 (1992).
77.
[77] Ju, J. W. and Tseng, K. H.: Improved two-dimensional micromechanical theory for brittle solids with randomly located interacting microcracks. Int. J. Damage Mechanics, 4, 23-57 (1995).
78.
[78] Budiansky, Y.: On the elastic moduli of some heterogeneous material. J. Mech. Phys. Solids, 13, 223-227 (1965).
79.
[79] Christensen, R. M. and Lo, K. H.: Solutions for effective shear properties in three phase space and cylinder model. J. Mech. Phys. Solids, 27, 315-330 (1979).
80.
[80] Hori, M. and Nemat-Nasser, S.: Double-inclusion model and overall moduli of multi-phase composites. Mech. Mater, 14, 189-206 (1993).
81.
[81] Huang, Y., Hu, K. X., and Chandra, A. A.: A generalized self-consistent mechanics method for composite materials with multiphace inclusions. J. Mech. Phys. Solids, 94, 491-504 (1994).
82.
[82] Zheng, Q.-S. and Du, D.-X.: An explicit and universally applicable estimate for the effective properties of multiphase composites which accounts for inclusion distribution. J. Mech. Phys. Solids, 49, 2265-2788 (2001).
83.
[83] Norris, A. N., Callegari, A. J., and Sheng, P A.: A generalized differential effective medium theory. J. Mech. Phys. Solids, 33, 525-543 (1985).
84.
[84] Buryachenko, V A. and Parton, V Z.: Multi-partical differential methods in the statics of composites. Priklad. Mekh. Tekhn. Fiz., 3, 148-156 (1992) (in Russian, Engl. Transl. J Applied Mech. Techn. Physics, 33, 455-464.).
85.
[85] Willis, J. R. (1978) Variational principles and bounds for the overall properties of composites. Continuum models of disordered systems (ed. J. W Provan), pp. 185-215. University of Waterloo Press, Waterloo.
86.
[86] Dyskin, A. V. and Muhilhaus, H. B.: Equilibrium bifurcations in dipole asymptotics model of periodic crack arrays. Continuum Modelsfor Materials with Microstructure, ed. H. B. Muhilhaus, pp. 69-104, John Wiley, New York, 1995.
87.
[87] Berveiller, M., Fassi-Fenri, O., and Hihi, A.: The problem of two plastic and heterogeneous inclusions in an anisotropic medium. International Journal of Engineering Science, 25, 691-709 (1987).
88.
[88] Rodin, G. J. and Hwang, Y L.: On the problem of linear elasticity for an infinite region containing a finite number of spherical inclusions. International Journal of Solids and Structures, 27, 145-159 (1991).
89.
[89] Moschovidis, Z. A. and Mura, T.: Two-ellipsoidal inhomogeneities by the equivalent inclusion method. ASME J. Appl. Mech., 42, 847-852 (1975).
90.
[90] Hori, M. and Nemat-Nasser, S.: Interacting microcracks near the tip in the process zone of a macrockrack. J. Mech. Phys. Solids, 35, 000-000 (1987).
91.
[91] Buryachenko, V A.: The overall elastoplastic behavior of multiphase matter. Acta Mech., 119, 93-117 (1996).
92.
[92] Buryachenko, V A.: Some nonlocal effects in graded random structure matrix composites. Mech. Res. Commun., 25, 117-122 (1998).
