A generalized differential effective medium theory.

*(English)*Zbl 0579.73114Summary: A generalization of the differential effective medium approximation (DEM) is discussed. The new scheme is applied to the estimation of the effective permittivity of a two phase dielectric composite. Ordinary DEM corresponds to a realizable microgeometry in which the composite is built up incrementally through a process of homogenization, with one phase always in dilute suspension and the other phase associated with the percolating backbone. The generalization of DEM assumes a third phase which acts as a backbone. The other two phases are progressively added to the backbone such that each addition is in an effectively homogeneous medium. A canonical ordinary differential equation is derived which describes the change in material properties as a function of the volume concentration \(\phi\) of the added phases in the composite. As \(\phi\) \(\to 1\), the effective medium approximation (EMA) is obtained. For \(\phi <1\), the result depends upon the backbone and the mixture path that is followed. The approach to EMA for \(\phi\) \(\cong 1\) is analysed and a generalization of Archie’s law for conductor-insulator composites is described. The conductivity mimics EMA above the percolation threshold and DEM as the conducting phase vanishes.

##### MSC:

74F15 | Electromagnetic effects in solid mechanics |

74E05 | Inhomogeneity in solid mechanics |

74A40 | Random materials and composite materials |

##### Keywords:

variable volume process; spherical grains; percolation properties; generalization of the differential effective medium approximation; estimation of the effective permittivity; two phase dielectric composite; homogenization; third phase which acts as a backbone; other two phases are progressively added to the backbone; effectively homogeneous medium; canonical ordinary differential equation; change in material properties; generalization of Archie’s law for conductor-insulator composites
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\textit{A. N. Norris} et al., J. Mech. Phys. Solids 33, 525--543 (1985; Zbl 0579.73114)

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##### References:

[1] | Bruggeman, D.A., Ann phyzik (Leipzig), 24, 636, (1935) |

[2] | Hashin, Z., J. appl. mech., 29, 143, (1962) |

[3] | Hashin, Z., J. composite mat., 2, 284, (1968) |

[4] | Hashin, Z.; Shtrikman, S., J. mech. phys. solids, 10, 335, (1962) |

[5] | Hori, M.; yonezawa, F., J. math. phys., 16, 352, (1975) |

[6] | Landau, L.D.; Lifshitz, E.M., Electrodynamics of continuous media, (1960), Pergamon Press Oxford · Zbl 0122.45002 |

[7] | Landauer, R., J. appl. phys., 23, 779, (1952) |

[8] | Landauer, R., Electrical transport and optical properties of inhomogeneous media, () · Zbl 1160.68305 |

[9] | Laws, N.; Walpole, L.J., () |

[10] | McGlaughlin, R., Int. J. engng sci., 15, 237, (1977) |

[11] | Mendelson, K.S.; Cohen, M.H., Geophysics, 47, 257, (1982) |

[12] | Milton, G.W., Phys. rev. lett., 46, 542, (1981) |

[13] | Milton, G.W., Physics and chemistry of porous media, () · Zbl 0624.41017 |

[14] | Norris, A.N., (), (to appear) |

[15] | Polder, D.; Van Sanies, J.M., Physica, 12, 257, (1946) |

[16] | Roscoe, R., Br. J. appl. phys., 3, 267, (1952) |

[17] | Sen, P.N.; Scala, C.; Cohen, M.H., Geophysics, 46, 781, (1981) |

[18] | Sheng, P.; Callegari, A.J., Appl. phys. lett., 44, 738, (1984) |

[19] | Stroud, D., Phys. rev., 12B, 3368, (1975) |

[20] | Yonezawa, F.; Cohen, M.H., J. appl. phys., 54, 2895, (1983) |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.