Key Words: average behavior, upper and lower bounds, Mori-Tanaka theory, composites, polycrystals

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1 Key Words: average behavior, upper and lower bounds, Mori-Tanaka theory, composites, polycrystals

2 (Q)1=C1(E)1, (0)2=C2(E)2 (4) QQQ= fi(o)1+f2(2+(1-fi-f2)(o)m E=flIE1+12(6)2+(1-fl-f2)(E)D (5a) (5b) (E)i=1E/D+-y, 2=1,2 (6) {0}i=Ci{1ED+ii} (7) QM=CMEM, 01=C1i, Q2=C2E2 (1) (o)i=cm{1e/d+li-ei} (g) Eti={CM-(CM-Ca)Si}-1(CM-C2)(E)D (12) (Q)M=CM{E}D (3) (o)i=c11e/d+cisiea (10) 1/z=CM(E)D+CM(Sz-I)Es (11) Ei={CM-(CM-Cz)Sti}-1(CM-C)CMl(Q)M E=(E)D+flit+f2Y2 (14)

3 E=CMl(o)M+flSlE1+f2S2E2 (15) (O)i=(Q)M+CM(Sti-I)Ei (16) A=CM+f1CM(S1-I){CM BEI+f151{CM C=AB-1 (17) -(CM-C1)S1}(CM-C1) +f2cm(s2-i){cm -(CM-C2)52F(CM-C2) -(CM-C1)S1}(CM-C1) +12S2{CM -(CM-C2)S2}-1(CM-C2) (18a) (18b) AK=kM(1-fi-f2) BK=1-fi-f2 Au=12M(1-fi-f2) Bu=1-fi-f2 k=flk1+f2k2 (23)

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8 materials, J. Mech. Phys. Solids, Vol. 11, pp , ) Walsh, J. B., Brace, W. F. and England, A. W.: Effect of porosity on compressibility of glass, Journal of the American Ceramic Society, Vol. 48, No. 12, pp , ) Coble, R. L. and Kingery, W. D.: Effect of porosity NCTAM 2000, pp , ) Omine, K., Ochiai, H. and Yoshida, N.: Estimation of in-situ strength of cement-treated soils based on a two-phase mixture model, Soils and Foundations, Vol. 38, No. 4, pp , ) Eshelby, J. D.: The determination of the elastic field of an ellipsoidal inclusion, and related problems, Proc. Roy. Soc. London, Vol. A241, pp , ) Mori, T. and Tanaka, K.: Average stress in matrix and average energy of materials with misfitting inclusions, Act. Metall., Vol. 21, pp , pp , ) Hashin, Z. and Shtrikman S.: A variational approach to the theory of the elastic behaviour of multiphase on physical properties of sintered alumina, Journal of the American Ceramic Society, Vol. 39, No. 11, pp , ) Nemat-Nasser, S., Iwakuma, T. and Hejazi, M.: On composites with periodic structure, Mechanics of Materials, Vol. 1, pp , ) Mura, T.: Micrornecharaics of Defects in Solids, Martinus Nijhoff Publ, ) Nishimatsu, C. and Gurland, J.: Experimental survey of the deformation of the hard-ductile two-phase alloy system WC-Co, Transaction of the American Society for Metals, Vol. 52, pp , AVERAGE CHARACTERISTICS OF COMPOSITES AND POLYCRYSTALS Shigeru KOYAMA, Tetsuo IWAKUMA, Tomoaki IWASAKI, Takao OGURA and Yasushi MITSUI An averaging method, Mori and Tanaka's theory, predicts the upper and lower bounds by Hashin and Shtrikman by exchanging material properties of the matrix and inclusion. However experimental results of porous media are very close to the upper bounds, and are also consistent with a numerical result of a body with periodic micro-structures. Here we propose a new approach using Mori and Tanaka's theory in which two materials are included into the matrix but the volume fraction of the matrix material is taken to be zero as a limit. Results show that either upper or lower bound by Hashin and Shtrikman is a probable estimate when the volume fraction of one material is very small.

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