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**The effect of spatial distribution on the effective behavior of composite materials and cracked media.**
*(English)*
Zbl 0919.73061

Summary: Estimates of the Hashin-Shtrikman type are developed for the overall moduli of composites consisting of a matrix containing one or more populations of inclusions, when the spatial correlations of inclusion locations take particular “ellipsoidal” forms. Inclusion shapes can be selected independently of the shapes adopted for the spatial correlations. The formulae that result are completely explicit and easy to use. They are likely to be useful, in particular, for composites that have undergone a prior macroscopically uniform large deformation. To the extent that the statistics that are assumed may not be realized exactly, the formulae provide approximations. Since, however, they are derived as variational approximations for composites with some explicit statistics that are realizable, they are free from some of the drawbacks of competitor approximations, which can generate tensors of effective moduli which fail to satisfy a necessary symmetry requirement. The new formulae are also the only ones known that take explicit account, at least approximately, of inclusion shape and spatial distribution independently.

### MSC:

74E30 | Composite and mixture properties |

### Keywords:

estimates of Hashin-Shtrikman type; macroscopically uniform large deformation; variational approximations
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\textit{P. Ponte Castañeda} and \textit{J. R. Willis}, J. Mech. Phys. Solids 43, No. 12, 1919--1951 (1995; Zbl 0919.73061)

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### References:

[1] | Benveniste, Y., A new approach to the application of Mori-Tanaka’s theory in composite materials, Mech. Mater., 6, 147-157 (1987) |

[2] | Budiansky, B.; O’Connell, R., Elastic moduli of a cracked solid, Int. J. Solids Struct., 12, 81-97 (1976) · Zbl 0318.73065 |

[3] | Eshelby, J. D., The determination of the elastic field of an ellipsoidal inclusion, and related problems, (Proc. R. Soc. Lond. A, 241 (1957)), 376-396 · Zbl 0079.39606 |

[4] | Hashin, Z., The differential scheme and its application to cracked materials, J. Mech. Phys. Solids, 36, 719-734 (1988) · Zbl 0673.73074 |

[5] | Hashin, Z.; Shtrikman, S., On some variational principles in anisotropic and nonhomogeneous elasticity, J. Mech. Phys. Solids, 10, 335-342 (1962) · Zbl 0111.41401 |

[6] | Hashin, Z.; Shtrikman, S., A variational approach to the theory of the elastic behavior of multiphase materials, J. Mech. Phys. Solids, 11, 127-140 (1963) · Zbl 0108.36902 |

[7] | Hill, R., The elastic behavior of a crystalline aggregate, (Proc. Phys. Soc. A, 65 (1952)), 349-354 |

[8] | Hill, R., Elastic properties of reinforced solids : some theoretical principles, J. Mech. Phys. Solids, 11, 357 (1963) · Zbl 0114.15804 |

[9] | Hill, R., Continuum micromechanics of elastoplastic polycrystals, J. Mech. Phys. Solids, 13, 89-101 (1965) · Zbl 0127.15302 |

[10] | Hoenig, A., Elastic moduli of a non-randomly cracked solid, Int. J. Solids Struct., 15, 137-154 (1979) · Zbl 0391.73089 |

[11] | Milton, G. W.; Kohn, R. V., Variational bounds on the effective moduli of anisotropic composites, J. Mech. Phys. Solids, 36, 597-629 (1988) · Zbl 0672.73012 |

[12] | Mori, T.; Tanaka, K., Average stress in matrix and average elastic energy of materials with misfitting inclusions, Acta Metall., 21, 571-574 (1973) |

[13] | Ponte, Castañeda P., The effective mechanical properties of nonlinear isotropic solids, J. Mech. Phys. Solids, 39, 45-71 (1991) · Zbl 0734.73052 |

[14] | Ponte Castañeda, P.; Willis, J. R., On the overall properties of nonlinearly viscous composites, (Proc. R. Soc. Lond. A, 416 (1988)), 217-244 · Zbl 0635.73006 |

[15] | Ponte Castañeda, P.; Zaidman, M., Constitutive models for porous materials with evolving microstructures, J. Mech. Phys. Solids, 42, 1459-1497 (1994) · Zbl 0823.73004 |

[16] | Stratonovich, R. L., (Topics in Theory of Random Noise, Vol. 1 (1963), Gordon and Breach: Gordon and Breach New York) |

[17] | Talbot, D. R.S.; Willis, J. R., Variational principles for nonlinear inhomogeneous media, IMA J. Appl. Math., 35, 39-54 (1985) · Zbl 0588.73025 |

[18] | Tandon, G. P.; Weng, G. J., Average stress in the matrix and effective moduli of randomly oriented composites, Comp. Sci. Technol., 27, 111-132 (1986) |

[19] | Taya, M.; Mura, T., On the stiffness and strength of aligned short-fiber reinforced composites containing fiber end cracks under uniaxial applied stress, J. Appl. Mech., 48, 361 (1981) · Zbl 0472.73115 |

[20] | Torquato, S., Random heterogeneous media : microstructure and improved bounds on effective properties, Appl. Mech. Rev., 44, 37-76 (1991) |

[21] | Walpole, L. J., On bounds for the overall elastic moduli of inhomogeneous systems: I, J. Mech. Phys. Solids, 14, 151-162 (1966) · Zbl 0139.18701 |

[22] | Walpole, L. J., On the overall elastic moduli of composite materials, J. Mech. Phys. Solids, 17, 235-251 (1969) · Zbl 0177.53204 |

[23] | Walsh, J. B., The effect of cracks on the compressibility of rocks, J. Geophys. Rev., 70, 2, 381-389 (1965) · Zbl 0136.22104 |

[24] | Weng, G. J., Some elastic properties of reinforced solids, with special reference to isotropic ones containing spherical inclusions, Int. J. Engng Sci., 22, 845-856 (1984) · Zbl 0556.73074 |

[25] | 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) · Zbl 0719.73004 |

[26] | Weng, G. J., Explicit evaluation of Willis’ bounds with ellipsoidal inclusions, Int. J. Engng Sci., 30, 83-92 (1992) · Zbl 0850.73028 |

[27] | Willis, J. R., Bounds and self-consistent estimates for the overall moduli of anisotropic composites, J. Mech. Phys. Solids, 25, 185-202 (1977) · Zbl 0363.73014 |

[28] | Willis, J. R., Variational principles and bounds for the overall properties of composites, (Provan, J. W., Continuum Models for Discrete Systems (1978), University of Waterloo Press: University of Waterloo Press Waterloo), 185-215 · Zbl 0363.73014 |

[29] | Willis, J. R., A polarization approach to the scattering of elastic waves—II. Multiple scattering from inclusions, J. Mech. Phys. Solids, 28, 307-327 (1980) · Zbl 0461.73013 |

[30] | Willis, J. R., Variational and related methods for the overall properties of composites, (Yih, C., Advances of Applied Mechanics, Volume 21 (1981), Academic Press: Academic Press New York), 1-78 · Zbl 0476.73053 |

[31] | Willis, J. R., The overall response of composite materials, ASME J. Appl. Mech., 50, 1202-1209 (1983) · Zbl 0539.73003 |

[32] | Willis, J. R., Some remarks on the application of the QCA to the determination of the overall response of a matrix/inclusion composite, J. Math. Phys., 25, 2116-2120 (1984) |

[33] | Wu, T. T., The effect of inclusions shape on the elastic moduli of a two-phase material, Int. J. Solids Struct., 2, 1-8 (1966) |

[34] | Zimmerman, R. W., The effect of microcracks on the elastic moduli of brittle materials, J. Mater. Sci. Lett., 4, 1457-1460 (1985) |

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