We consider a linearly thermoelastic composite medium, which consists of a homogeneous matrix containing a statistically homogeneous set of ellipsoidal inclusions and subjected to inhomogeneous boundary conditions. We use the multiparticle effective field method (MEFM) based on the theory of functions of random variables and Green's functions; for references, see Buryachenko, V.A. Applied Mechanics Reviews, 54, 1-47 (2001). Within this method, we derive a hierarchy of statistical moment equations for conditional averages of the stresses in the inclusions. The hierarchy is established by introducing the notion of an effective field. In this way the interaction of different inclusions is taken directly into account in the framework of the homogeneity hypothesis of the effective field. The non-local integral equation for the statistical average of stresses inside the inclusions is solved by three different methods: the quadrature method, the iteration method, and the Fourier transform method with subsequent comparative analysis. The standard scheme of iteration and Fourier transform methods permit us to obtain the explicit representations for the non-local integral and differential operators, respectively, of any order describing overall effective properties as well as the stress concentration factor in the components. It is shown that the integral operator reduces to the differential one for sufficiently smooth statistical average stress fields. With some additional assumptions, the proposed method is reduced to the perturbation method as well as to the “quasi-crystalline” approach. For some concrete numerical examples we can demonstrate the advantage of the quadrature method and the iteration method over the Fourier transform method.
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References
1.
[1] Christensen, R. M.: Mechanics of Composite Materials, Wiley Interscience, New York , 1979.
2.
[2] Willis, J. R.: Variational and related methods for the overall properties of composites . Advances in Applied Mechanics, 21, 1-78 (1981).
3.
[3] Mura, T.: Micromechanics of Defects in Solids, Martinus Nijhoff, Dordrecht , 1987.
4.
[4] Nemat-Nasser, S., and Hori, M.: Micromechanics: Overall Properties of Heterogeneous Materials, Elsevier, Amsterdam , 1993.
5.
[5] Markov, K. Z.: Elementary micromechanics of heterogeneous media. Heterogeneous Media: Modelling and Simulation, pp. 1-168, eds., K. Z. Markov and L. Preziosi, Birkhäuser, Boston , 1999.
6.
[6] Kanaun, S. K., and Levin, V. M.: Effective field method in mechanics of matrix composite materials. Advances in Mathematical Modeling of Composite Materials, pp. 1-58, ed., K. Z. Markov, World Scientific, Singapore , 1994.
7.
[7] Buryachenko, V. A.: Multiparticle effective field and related methods in micromechanics of composite materials . Appl. Mech. Reviews, 54, 1-47 (2001).
8.
[8] Torquato, S.: Random Heterogeneous Materials: Microstucture and Macroscopic Properties, Springer-Verlag, Berlin , 2002.
9.
[9] Carol, I., and Bazant, Z. P.: Damage and plasticity in microplane theory . Int. J. Solids Structures, 34, 3807-3835 (1997).
10.
[10] Fleck, N. A., and Hutchinson, J. W.: Strain gradient plasticity. Advances in Applied Mechanics, Vol. 33, pp. 295-361, eds., J.W. Hutchinson & T.Y. Wu, Academic, New York , 1997.
11.
[11] Eringen, A. C.: Micromedium Field Theories I. Foundations and Solids, Springer, Berlin , 1999.
12.
[12] Aifantis E. C.: Gradient deformation models at nano, micro, and macro scales . ASME. J Eng. Mater. Technol., 121, 189-202 (1999).
13.
[13] Ganghoffer, J. F., and de Borst R.: A new framework in non-local mechanics . Int. J. Eng. Sci., 38, 453-486 (2000).
14.
[14] Picu, R. C.: On the functional form of non-local elasticity kernels . J. Mech. Phys. Solids, 50, 1923-1939 (2002).
15.
[15] Kröner E.: Elasticity theory of materials with long range cohesive forces . Int. J. Solid Structures, 3, 731-342 (1967).
16.
[16] Kröner E.: Statistical Continuum Mechanics, Springer-Verlag, Wien , 1972.
17.
[17] Kunin, I. A., and Vaisman, A. M.: On problems of the non-local theory of elasticity. Fundamental Aspects of Dislocation Theory, pp. 747-757, eds., J. Simmons, R. de Wit & R. Bullough, US National Bureau of Standards, Washington DC , 1970.
18.
[18] Eringen, A. C., and Edelen, D. G. B.: On non-local elasticity . Int. J. Eng. Sci., 10, 233-243 (1972).
19.
[19] Beran, M. J., and McCoy, J. J.: Mean field variations in a statistical sample of heterogeneous linearly elastic solids . Int. J. Solid Structures, 6, 1035-1054 (1970).
20.
Engl. Transl. J. Appl. Math. Mech., 35, 694-701 (1971).
21.
[21] Lax, M.: Multiple scattering of waves II. The effective fields dense systems . Phys. Rev., 85, 621-629 (1952).
22.
(in Russian). Engl. Transl. Int. Appl. Mech., 32, 341-348 (1996).
23.
[23] Drugan, W. J., and Willis, J. R.: A micromechanics-based non-local constitutive equation and estimates of representative volume elements for elastic composites . J. Mech. Phys. Solids, 44, 497-524 (1996).
24.
[24] Drugan, W. J.: Micromechanics-based variational estimations for a higher-order non-local constitutive equation and optimal choice of effective moduli for elastic composites . J. Mech. Phys. Solids, 48, 1359-1387 (2000).
25.
[25] Luciano, R., and Willis, J. R.: Bounds on non-local effective relations for random composites loaded by configuration-dependent body force . J. Mech. Phys. Solids, 48, 1827-1849 (2000).
26.
[26] Luciano, R., and Willis, J. R.: Non-local effective relations for fibre-reinforced composites loaded by configuration-dependent body forces . J. Mech. Phys. Solids, 49, 2705-2717 (2001).
27.
[27] Willis, J. R.: The non-local influence of density variations in a composite . Int. J. Solids Structures, 21, 805-817 (1985).
28.
[28] Buryachenko, V. A., and Lipanov, A. M.: Thermoelastic stress concentration at ellipsoidal inclusions in matrix composites in the region of strongly varying external stress and temperature fields. Deformation and Fracture of Structural-Inhomogeneous Materials, pp. 12-19, eds., O. B. Naimark & S. E. Evlampieva, AN SSSR, Sverdlovsk , 1992 (in Russian).
29.
[29] Buryachenko, V. A.: Some non-local effects in graded random structure matrix composites . Mech. Res. Commun., 25, 117-122 (1998).
30.
[30] Buryachenko, V. A. and Rammerstorfer, F. G.: Micromechanics and non-local effects in graded random structure matrix composites. IUTAM Symp. on Transformation Problems in Composite and Active Materials, pp. 197-206, eds., Y.A. Bahei-El-Din & G. J. Dvorak, Kluwer Academic, Dordrecht , 1998.
31.
[31] Buryachenko, V. A.: Effective thermoelastic properties of graded doubly periodic particulate composites in varying external stress fields . Int. J. Solids Structures, 36, 3861-3885 (1999).
32.
[32] Bakhvalov, N. G., and Panasenko, G.: Homogenization: Averaging Processes in Periodic Media, Kluwer, Dordrecht , 1989.
33.
[33] Kalamkarov, A. L., and Kolpakov, A. G.: Analysis, Design and Optimization of Composite Shells, Wiley, New York , 1997.
34.
[34] Gambin, B., and Kröner, E.: Higher-order terms in the homogenized stress-strain relation of periodic elastic media . Phys. Status Solidi, b151, 513-530 (1989).
35.
[35] Boutin, C.: Microstructural effects in elastic composites . Int. J. Solids Structures, 33, 1023-1051 (1996).
36.
[36] Forest, S., and Sab, K.: Cosserat overall modeling of heterogeneous materials . Mech. Res. Commun., 25, 449-454 (1998).
37.
[37] Smyshlyaev, V. P., and Cherednichenko, K. D.: A rigorous derivation of strain gradient effects in the overall behavior of periodic heterogeneous media . J. Mech. Phys. Solids, 48, 1325-1357 , (2000).
38.
[38] Bouyge, F., Jasiuk, I., and Ostoja-Starzewski, M.: A micromechanically based couple-stress model of an elastic two-phase composite . Int. J. Solids Structures, 38, 1721-1735 (2001).
39.
[39] Pagano, N.J., and Yuan, F. G.: Significance of effective modulus theory (homogenization), in composite laminate mechanics . Compos. Sci. Technol., 60, 2471-2488 (2000).
40.
Engl. Transl. Mech. Solids, 31(3), 2-11 (1996).
41.
[41] Lekhnitskii, A. G.: Theory of Elasticity of an Anisotropic Elastic Body, Holder Day, San Francisco , 1963.
42.
[42] Kröner, E.: Bounds for effective moduli of disordered materials , J. Mech. Phys. Solids, 25, 137-155 (1977).
43.
[43] Eshelby, J. D.: The determination of the elastic field of an ellipsoidal inclusion, and related problems . Proc. R. Soc. London, A 241, 376-396 (1957).
44.
[44] Dvorak, G. J., and Benveniste, Y.: On transformation strains and uniform fields in multiphase elastic media . Proc. R. Soc. London, A 437, 291-310 (1992).
45.
[45] Lee, J., and Mal, A.: A volume integral equation technique for multiple inclusion and crack interaction problems . J. Appl. Mech. - Trans. ASME, 64, 23-31 (1997).
46.
[46] Lee, J., and Mal, A.: Characterization of matrix damage in metal matrix composites under transverse loads . Computational Mech., 21, 339-346 (1998).
47.
[47] Delves, L. M., and Mohamed, J. L.: Computational Methods for Integral Equations, Cambridge University Press, Cambridge , 1985.
48.
[48] Varga, R. S.: Matrix Iterative Analysis, Springer, Berlin , 2000.
49.
[49] Pipkin, A. C.: A Course on Integral Equations, Springer, New York , 1991.
50.
[50] Shilov, G. E.: Generalized Functions and Partial Differential Equations, Gordon & Breach, New York , 1968.
51.
[51] Treves, F.: Introduction to Pseudodifferential and Fourier Integral Operators, Vol. 1, Plenum Press, New York , 1980.
52.
[52] Zuiker, J. R., and Dvorak, G. J.: The effective properties of functionally graded composites-I. Extension of the Mori-Tanaka method to linearly varying fields . Compos. Eng., 4, 19-35 (1994).
53.
[53] Smyshlyaev, V. P., and Fleck, N. A.: The role of strain gradients in the grain size effects for polycrystals . J. Mech. Phys. Solids, 44, 465-495 (1996).
54.
[54] Torquato, S., and Lado, F.: Improved bounds on the effective elastic moduli of random arrays of cylinders . ASME. J. Appl. Mech., 59, 1-6 (1992).
55.
[55] Hansen, J. P., and McDonald, I. R.: Theory of Simple Liquids, Academic, New York , 1986.
56.
[56] Buryachenko, V. A., Pagano, N. J., Kim, R. Y., and Spowart, J. E.: Quantitative description of random microstructures of composites and their effective elastic moduli . Int. J. Solids Structures, 40, in press (2003).
57.
[57] Lee, J. A. and Mykkanen, D. L.: Metal and Polymer Matrix Composites. Noyes Data Corporation, New York , 1987.
58.
[58] Zuiker, J. R.: Functionally graded materials: Choice of micromechanics model and limitations in property variation . Compos. Eng., 5, 807-819 (1995).
59.
[59] Reiter, T., Dvorak, G. J., and Tvergard, V.: Micromechanical models for graded composite materials . J. Mech. Phys. Solids, 45, 1281-1302 (1997).
